Theorem equsal 1960

Description: A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 3-Oct-2016.)

Hypotheses

Ref	Expression
equsal.1	⊢ Ⅎxψ
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 ⊢ Ⅎxψ
3	2	19.3 1785	. . . . 5 ⊢ (∀xψ ↔ ψ)
4	1, 3	syl6bbr 254	. . . 4 ⊢ (x = y → (φ ↔ ∀xψ))
5	4	pm5.74i 236	. . 3 ⊢ ((x = y → φ) ↔ (x = y → ∀xψ))
6	5	albii 1566	. 2 ⊢ (∀x(x = y → φ) ↔ ∀x(x = y → ∀xψ))
7	2	nfri 1762	. . . . 5 ⊢ (ψ → ∀xψ)
8	7	a1d 22	. . . 4 ⊢ (ψ → (x = y → ∀xψ))
9	2, 8	alrimi 1765	. . 3 ⊢ (ψ → ∀x(x = y → ∀xψ))
10		ax9o 1950	. . 3 ⊢ (∀x(x = y → ∀xψ) → ψ)
11	9, 10	impbii 180	. 2 ⊢ (ψ ↔ ∀x(x = y → ∀xψ))
12	6, 11	bitr4i 243	1 ⊢ (∀x(x = y → φ) ↔ ψ)

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  equsalh  1961  equsex  1962  dvelimf  1997  sb6x  2029  fun11  5160