Theorem chvar 1688

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 143	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	1, 3	spim 1674	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
5		chvar.3	. 2 ⊢ 𝜑
6	4, 5	mpg 1386	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 104  Ⅎwnf 1395

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-4 1446  ax-i9 1469  ax-ial 1473

This theorem depends on definitions:  df-bi 116  df-nf 1396

This theorem is referenced by:  csbhypf  2969  opelopabsb  4098  findes  4433  fvmptssdm  5402  dfoprab4f  5979  dom2lem  6545  uzind4s  9141  fsumsplitf  10865