Theorem spimv1 2238

Description: Version of spim 2351 with a disjoint variable condition, which does not require ax-13 2333. See spimvw 2045 for a version with two disjoint variable conditions, requiring fewer axioms, and spimv 2354 for another variant. (Contributed by BJ, 31-May-2019.)

Hypotheses

Ref	Expression
spimv1.nf	⊢ Ⅎ𝑥𝜓
spimv1.1	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
spimv1	⊢ (∀𝑥𝜑 → 𝜓)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem spimv1

Step	Hyp	Ref	Expression
1		spimv1.nf	. 2 ⊢ Ⅎ𝑥𝜓
2		ax6ev 2023	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
3		spimv1.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	2, 3	eximii 1880	. 2 ⊢ ∃𝑥(𝜑 → 𝜓)
5	1, 4	19.36i 2216	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1599  Ⅎwnf 1827

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-12 2162

This theorem depends on definitions:  df-bi 199  df-ex 1824  df-nf 1828

This theorem is referenced by:  cbv3v2  2239  cbv3v  2309  bj-chvarv  33327