Theorem spimv1 2115

Description: Version of spim 2254 with a dv condition, which does not require ax-13 2246. See spimvw 1927 for a version with two dv conditions, requiring fewer axioms, and spimv 2257 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 1890	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
3		spimv1.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	2, 3	eximii 1764	. 2 ⊢ ∃𝑥(𝜑 → 𝜓)
5	1, 4	19.36i 2099	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1481  Ⅎwnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-nf 1710

This theorem is referenced by:  cbv3v  2172  bj-chvarv  32725  bj-cbv3v2  32727  wl-cbv3vv  33307