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Metamath Proof Explorer

Theorem 2mosOLD

Description: Obsolete version of 2mos as of 21-May-2025. (Contributed by NM, 10-Feb-2005) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypothesis 2mos.1 ⊢ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → ( 𝜑 ↔ 𝜓 ) )
Assertion 2mosOLD ( ∃ 𝑧 ∃ 𝑤 ∀ 𝑥 ∀ 𝑦 ( 𝜑 → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) )

Proof

Step Hyp Ref Expression
1 2mos.1 ⊢ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → ( 𝜑 ↔ 𝜓 ) )
2 2mo ⊢ ( ∃ 𝑧 ∃ 𝑤 ∀ 𝑥 ∀ 𝑦 ( 𝜑 → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) )
3 nfv ⊢ Ⅎ 𝑦 𝑥 = 𝑧
4 3 sbrim ⊢ ( [ 𝑤 / 𝑦 ] ( 𝑥 = 𝑧 → 𝜑 ) ↔ ( 𝑥 = 𝑧 → [ 𝑤 / 𝑦 ] 𝜑 ) )
5 1 expcom ⊢ ( 𝑦 = 𝑤 → ( 𝑥 = 𝑧 → ( 𝜑 ↔ 𝜓 ) ) )
6 5 pm5.74d ⊢ ( 𝑦 = 𝑤 → ( ( 𝑥 = 𝑧 → 𝜑 ) ↔ ( 𝑥 = 𝑧 → 𝜓 ) ) )
7 6 sbievw ⊢ ( [ 𝑤 / 𝑦 ] ( 𝑥 = 𝑧 → 𝜑 ) ↔ ( 𝑥 = 𝑧 → 𝜓 ) )
8 4 7 bitr3i ⊢ ( ( 𝑥 = 𝑧 → [ 𝑤 / 𝑦 ] 𝜑 ) ↔ ( 𝑥 = 𝑧 → 𝜓 ) )
9 8 pm5.74ri ⊢ ( 𝑥 = 𝑧 → ( [ 𝑤 / 𝑦 ] 𝜑 ↔ 𝜓 ) )
10 9 sbievw ⊢ ( [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 ↔ 𝜓 )
11 10 anbi2i ⊢ ( ( 𝜑 ∧ [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 ) ↔ ( 𝜑 ∧ 𝜓 ) )
12 11 imbi1i ⊢ ( ( ( 𝜑 ∧ [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) )
13 12 2albii ⊢ ( ∀ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) ↔ ∀ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) )
14 13 2albii ⊢ ( ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) )
15 2 14 bitri ⊢ ( ∃ 𝑧 ∃ 𝑤 ∀ 𝑥 ∀ 𝑦 ( 𝜑 → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) )