Theorem bj-spimev 33218

Description: Version of spime 2396 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-spimev.1	⊢ Ⅎ𝑥𝜑
bj-spimev.2	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
bj-spimev	⊢ (𝜑 → ∃𝑥𝜓)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem bj-spimev

Step	Hyp	Ref	Expression
1		bj-spimev.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2	1	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
3		bj-spimev.2	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	2, 3	bj-spimedv 33217	. 2 ⊢ (⊤ → (𝜑 → ∃𝑥𝜓))
5	4	mptru 1661	1 ⊢ (𝜑 → ∃𝑥𝜓)

Syntax hints:   → wi 4  ⊤wtru 1654  ∃wex 1875  Ⅎwnf 1879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-12 2213

This theorem depends on definitions:  df-bi 199  df-tru 1657  df-ex 1876  df-nf 1880

This theorem is referenced by:  bj-spimevv  33220