Theorem bj-spimevv 32722

Description: Version of spimev 2259 with a dv condition, which does not require ax-13 2246. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-spimevv.1	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
bj-spimevv	⊢ (𝜑 → ∃𝑥𝜓)

Distinct variable groups:   𝑥,𝑦   𝜑,𝑥

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

Proof of Theorem bj-spimevv

Step	Hyp	Ref	Expression
1		nfv 1843	. 2 ⊢ Ⅎ𝑥𝜑
2		bj-spimevv.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
3	1, 2	bj-spimev 32720	1 ⊢ (𝜑 → ∃𝑥𝜓)

Syntax hints:   → wi 4  ∃wex 1704

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-tru 1486  df-ex 1705  df-nf 1710

This theorem is referenced by:  bj-axsep  32793  bj-dtru  32797