Theorem spimefv 2198

Description: Version of spime 2407 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by BJ, 31-May-2019.)

Hypotheses

Ref	Expression
spimefv.1	⊢ Ⅎ𝑥𝜑
spimefv.2	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
spimefv	⊢ (𝜑 → ∃𝑥𝜓)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem spimefv

Step	Hyp	Ref	Expression
1		spimefv.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2	1	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
3		spimefv.2	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	2, 3	spimedv 2197	. 2 ⊢ (⊤ → (𝜑 → ∃𝑥𝜓))
5	4	mptru 1544	1 ⊢ (𝜑 → ∃𝑥𝜓)

Syntax hints:   → wi 4  ⊤wtru 1538  ∃wex 1780  Ⅎwnf 1784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-tru 1540  df-ex 1781  df-nf 1785

This theorem is referenced by:  dtru  5271