Theorem chvarv 2013

Description: Implicit substitution of y for x into a theorem. (Contributed by NM, 20-Apr-1994.)

Hypotheses

Ref	Expression
chv.1	⊢ (x = y → (φ ↔ ψ))
chv.2	⊢ φ

Assertion

Ref	Expression
chvarv	⊢ ψ

Distinct variable group:   ψ,x

Allowed substitution hints:   φ(x,y)   ψ(y)

Proof of Theorem chvarv

Step	Hyp	Ref	Expression
1		chv.1	. . 3 ⊢ (x = y → (φ ↔ ψ))
2	1	spv 1998	. 2 ⊢ (∀xφ → ψ)
3		chv.2	. 2 ⊢ φ
4	2, 3	mpg 1548	1 ⊢ ψ

Syntax hints:   → wi 4   ↔ wb 176

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:  axext3  2336