Theorem equsal 1147

Description: A useful equivalence related to substitution.

Hypotheses

Ref	Expression
equsal.1	⊢ (ψ → ∀xψ)
equsal.2	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
equsal	⊢ (∀x(x = y → φ) ↔ ψ)

Proof of Theorem equsal

Step	Hyp	Ref	Expression
1		equsal.2	. . . . 5 ⊢ (x = y → (φ ↔ ψ))
2		equsal.1	. . . . . 6 ⊢ (ψ → ∀xψ)
3	2	19.3 1027	. . . . 5 ⊢ (∀xψ ↔ ψ)
4	1, 3	syl6bbr 536	. . . 4 ⊢ (x = y → (φ ↔ ∀xψ))
5	4	pm5.74i 582	. . 3 ⊢ ((x = y → φ) ↔ (x = y → ∀xψ))
6	5	albii 996	. 2 ⊢ (∀x(x = y → φ) ↔ ∀x(x = y → ∀xψ))
7		ax-1 4	. . . . 5 ⊢ (∀xψ → (x = y → ∀xψ))
8	7	a5i 986	. . . 4 ⊢ (∀xψ → ∀x(x = y → ∀xψ))
9	2, 8	syl 10	. . 3 ⊢ (ψ → ∀x(x = y → ∀xψ))
10		ax-9o 1119	. . 3 ⊢ (∀x(x = y → ∀xψ) → ψ)
11	9, 10	impbi 157	. 2 ⊢ (ψ ↔ ∀x(x = y → ∀xψ))
12	6, 11	bitr4 176	1 ⊢ (∀x(x = y → φ) ↔ ψ)

Syntax hints:   → wi 3   ↔ wb 146  ∀wal 951   = wceq 953

This theorem is referenced by:  equsex 1148  dvelimfALT 1149  fun11 3548

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-4 970  ax-5o 972  ax-9o 1119

This theorem depends on definitions:  df-bi 147  df-an 225

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