Theorem chvarfv 1688

Description: Implicit substitution of 𝑦 for 𝑥 into a theorem. Version of chvar 1745 with a disjoint variable condition. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by BJ, 31-May-2019.)

Hypotheses

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

Assertion

Ref	Expression
chvarfv	⊢ 𝜓

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem chvarfv

Step	Hyp	Ref	Expression
1		chvarfv.nf	. . 3 ⊢ Ⅎ𝑥𝜓
2		chvarfv.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	2	biimpd 143	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	1, 3	spimfv 1687	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
5		chvarfv.2	. 2 ⊢ 𝜑
6	4, 5	mpg 1439	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 104  Ⅎwnf 1448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-i9 1518  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-nf 1449

This theorem is referenced by:  fproddivapf  11572  fprodsplitf  11573