Theorem chvar 1771

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 144	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	1, 3	spim 1752	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
5		chvar.3	. 2 ⊢ 𝜑
6	4, 5	mpg 1465	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 105  Ⅎwnf 1474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-i9 1544  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-nf 1475

This theorem is referenced by:  csbhypf  3123  opelopabsb  4294  findes  4639  fvmptssdm  5646  dfoprab4f  6251  dom2lem  6831  uzind4s  9664  fsumsplitf  11573