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Metamath Proof Explorer

Theorem elabgtOLDOLD

Description: Obsolete version of elabgt as of 11-May-2025. (Contributed by NM, 7-Nov-2005) (Proof shortened by Andrew Salmon, 8-Jun-2011) Reduce axiom usage. (Revised by GG, 12-Oct-2024) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion elabgtOLDOLD ( ( 𝐴 ∈ 𝐵 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ) → ( 𝐴 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝜓 ) )

Proof

Step Hyp Ref Expression
1 elab6g ⊢ ( 𝐴 ∈ 𝐵 → ( 𝐴 ∈ { 𝑥 ∣ 𝜑 } ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) )
2 1 adantr ⊢ ( ( 𝐴 ∈ 𝐵 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ) → ( 𝐴 ∈ { 𝑥 ∣ 𝜑 } ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) )
3 elisset ⊢ ( 𝐴 ∈ 𝐵 → ∃ 𝑥 𝑥 = 𝐴 )
4 biimp ⊢ ( ( 𝜑 ↔ 𝜓 ) → ( 𝜑 → 𝜓 ) )
5 4 imim3i ⊢ ( ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) → ( ( 𝑥 = 𝐴 → 𝜑 ) → ( 𝑥 = 𝐴 → 𝜓 ) ) )
6 5 al2imi ⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) ) )
7 19.23v ⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 ) )
8 6 7 imbitrdi ⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) → ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 ) ) )
9 8 com3r ⊢ ( ∃ 𝑥 𝑥 = 𝐴 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) → 𝜓 ) ) )
10 biimpr ⊢ ( ( 𝜑 ↔ 𝜓 ) → ( 𝜓 → 𝜑 ) )
11 10 imim2i ⊢ ( ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) → ( 𝑥 = 𝐴 → ( 𝜓 → 𝜑 ) ) )
12 11 alimi ⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) → ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜓 → 𝜑 ) ) )
13 bi2.04 ⊢ ( ( 𝑥 = 𝐴 → ( 𝜓 → 𝜑 ) ) ↔ ( 𝜓 → ( 𝑥 = 𝐴 → 𝜑 ) ) )
14 13 albii ⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜓 → 𝜑 ) ) ↔ ∀ 𝑥 ( 𝜓 → ( 𝑥 = 𝐴 → 𝜑 ) ) )
15 19.21v ⊢ ( ∀ 𝑥 ( 𝜓 → ( 𝑥 = 𝐴 → 𝜑 ) ) ↔ ( 𝜓 → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) )
16 14 15 sylbb ⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜓 → 𝜑 ) ) → ( 𝜓 → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) )
17 12 16 syl ⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) → ( 𝜓 → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) )
18 17 a1i ⊢ ( ∃ 𝑥 𝑥 = 𝐴 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) → ( 𝜓 → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) ) )
19 9 18 impbidd ⊢ ( ∃ 𝑥 𝑥 = 𝐴 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 ) ) )
20 3 19 syl ⊢ ( 𝐴 ∈ 𝐵 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 ) ) )
21 20 imp ⊢ ( ( 𝐴 ∈ 𝐵 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ) → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 ) )
22 2 21 bitrd ⊢ ( ( 𝐴 ∈ 𝐵 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ) → ( 𝐴 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝜓 ) )