Theorem spimv1 2101

Description: Version of spim 2242 with a dv condition, which does not require ax-13 2234. See spimvw 1914 for a version with two dv conditions, requiring fewer axioms, and spimv 2245 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 1877	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
3		spimv1.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	2, 3	eximii 1754	. 2 ⊢ ∃𝑥(𝜑 → 𝜓)
5	1, 4	19.36i 2086	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1473  Ⅎwnf 1699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nf 1701

This theorem is referenced by:  cbv3v  2158  bj-chvarv  31912  bj-cbv3v2  31914  wl-cbv3vv  32486