Theorem chvar 1986

Description: Implicit substitution of y for x into a theorem. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro, 3-Oct-2016.)

Hypotheses

Ref	Expression
chvar.1	⊢ Ⅎxψ
chvar.2	⊢ (x = y → (φ ↔ ψ))
chvar.3	⊢ φ

Assertion

Ref	Expression
chvar	⊢ ψ

Proof of Theorem chvar

Step	Hyp	Ref	Expression
1		chvar.1	. . 3 ⊢ Ⅎxψ
2		chvar.2	. . . 4 ⊢ (x = y → (φ ↔ ψ))
3	2	biimpd 198	. . 3 ⊢ (x = y → (φ → ψ))
4	1, 3	spim 1975	. 2 ⊢ (∀xφ → ψ)
5		chvar.3	. 2 ⊢ φ
6	4, 5	mpg 1548	1 ⊢ ψ

Syntax hints:   → wi 4   ↔ wb 176  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:  csbhypf  3172  opelopabsb  4698