Theorem equsalh 1961

Description: A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.)

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.1	. . 3 ⊢ (ψ → ∀xψ)
2	1	nfi 1551	. 2 ⊢ Ⅎxψ
3		equsalh.2	. 2 ⊢ (x = y → (φ ↔ ψ))
4	2, 3	equsal 1960	1 ⊢ (∀x(x = y → φ) ↔ ψ)

This proof depends on syntax axioms:   → wi 4   ↔ wb 176  ∀wal 1540

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

This theorem is used by:  dvelimh  1964  dvelimALT  2133  dvelimf-o  2180