Theorem spimfv 1692

Description: Specialization, using implicit substitution. Version of spim 1731 with a disjoint variable condition. See spimv 1804 for another variant. (Contributed by NM, 10-Jan-1993.) (Revised by BJ, 31-May-2019.)

Hypotheses

Ref	Expression
spimfv.nf	⊢ Ⅎ𝑥𝜓
spimfv.1	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
spimfv	⊢ (∀𝑥𝜑 → 𝜓)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem spimfv

Step	Hyp	Ref	Expression
1		spimfv.nf	. 2 ⊢ Ⅎ𝑥𝜓
2		a9ev 1690	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
3		spimfv.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	2, 3	eximii 1595	. 2 ⊢ ∃𝑥(𝜑 → 𝜓)
5	1, 4	19.36i 1665	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1346  Ⅎwnf 1453

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-nf 1454

This theorem is referenced by:  chvarfv  1693