Theorem equsalh 1611

Description: A useful equivalence related to substitution. New proofs should use equsal 1612 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)

Hypotheses

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

Assertion

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

Proof of Theorem equsalh

Step	Hyp	Ref	Expression
1		equsalh.2	. . . . 5 ⊢ (x = y → (φ ↔ ψ))
2		equsalh.1	. . . . . 6 ⊢ (ψ → ∀xψ)
3	2	19.3h 1442	. . . . 5 ⊢ (∀xψ ↔ ψ)
4	1, 3	syl6bbr 187	. . . 4 ⊢ (x = y → (φ ↔ ∀xψ))
5	4	pm5.74i 169	. . 3 ⊢ ((x = y → φ) ↔ (x = y → ∀xψ))
6	5	albii 1356	. 2 ⊢ (∀x(x = y → φ) ↔ ∀x(x = y → ∀xψ))
7	2	a1d 22	. . . 4 ⊢ (ψ → (x = y → ∀xψ))
8	2, 7	alrimih 1355	. . 3 ⊢ (ψ → ∀x(x = y → ∀xψ))
9		ax9o 1585	. . 3 ⊢ (∀x(x = y → ∀xψ) → ψ)
10	8, 9	impbii 117	. 2 ⊢ (ψ ↔ ∀x(x = y → ∀xψ))
11	6, 10	bitr4i 176	1 ⊢ (∀x(x = y → φ) ↔ ψ)

Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1240

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-i9 1420  ax-ial 1424

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  sb6x  1659  dvelimfALT2  1695  dvelimALT  1883  dvelimfv  1884  dvelimor  1891