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Metamath Proof Explorer

Theorem pm13.192

Description: Theorem *13.192 in WhiteheadRussell p. 179. (Contributed by Andrew Salmon, 3-Jun-2011) (Revised by NM, 4-Jan-2017)

Ref Expression
Assertion pm13.192 ( ∃ 𝑦 ( ∀ 𝑥 ( 𝑥 = 𝐴 ↔ 𝑥 = 𝑦 ) ∧ 𝜑 ) ↔ [ 𝐴 / 𝑦 ] 𝜑 )

Proof

Step Hyp Ref Expression
1 biimpr ⊢ ( ( 𝑥 = 𝐴 ↔ 𝑥 = 𝑦 ) → ( 𝑥 = 𝑦 → 𝑥 = 𝐴 ) )
2 1 alimi ⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 ↔ 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝑥 = 𝐴 ) )
3 eqeq1 ⊢ ( 𝑥 = 𝑦 → ( 𝑥 = 𝐴 ↔ 𝑦 = 𝐴 ) )
4 3 equsalvw ⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝑥 = 𝐴 ) ↔ 𝑦 = 𝐴 )
5 2 4 sylib ⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 ↔ 𝑥 = 𝑦 ) → 𝑦 = 𝐴 )
6 eqeq2 ⊢ ( 𝐴 = 𝑦 → ( 𝑥 = 𝐴 ↔ 𝑥 = 𝑦 ) )
7 6 eqcoms ⊢ ( 𝑦 = 𝐴 → ( 𝑥 = 𝐴 ↔ 𝑥 = 𝑦 ) )
8 7 alrimiv ⊢ ( 𝑦 = 𝐴 → ∀ 𝑥 ( 𝑥 = 𝐴 ↔ 𝑥 = 𝑦 ) )
9 5 8 impbii ⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 ↔ 𝑥 = 𝑦 ) ↔ 𝑦 = 𝐴 )
10 9 anbi1i ⊢ ( ( ∀ 𝑥 ( 𝑥 = 𝐴 ↔ 𝑥 = 𝑦 ) ∧ 𝜑 ) ↔ ( 𝑦 = 𝐴 ∧ 𝜑 ) )
11 10 exbii ⊢ ( ∃ 𝑦 ( ∀ 𝑥 ( 𝑥 = 𝐴 ↔ 𝑥 = 𝑦 ) ∧ 𝜑 ) ↔ ∃ 𝑦 ( 𝑦 = 𝐴 ∧ 𝜑 ) )
12 sbc5 ⊢ ( [ 𝐴 / 𝑦 ] 𝜑 ↔ ∃ 𝑦 ( 𝑦 = 𝐴 ∧ 𝜑 ) )
13 11 12 bitr4i ⊢ ( ∃ 𝑦 ( ∀ 𝑥 ( 𝑥 = 𝐴 ↔ 𝑥 = 𝑦 ) ∧ 𝜑 ) ↔ [ 𝐴 / 𝑦 ] 𝜑 )