Theorem speivw 1977

Description: Version of spei 2412 with a disjoint variable condition, which does not require ax-13 2390 (neither ax-7 2015 nor ax-12 2177). (Contributed by BJ, 31-May-2019.)

Hypotheses

Ref	Expression
speivw.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
speivw.2	⊢ 𝜓

Assertion

Ref	Expression
speivw	⊢ ∃𝑥𝜑

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem speivw

Step	Hyp	Ref	Expression
1		speivw.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2	1	biimprd 250	. 2 ⊢ (𝑥 = 𝑦 → (𝜓 → 𝜑))
3		speivw.2	. 2 ⊢ 𝜓
4	2, 3	speiv 1976	1 ⊢ ∃𝑥𝜑

Syntax hints:   → wi 4   ↔ wb 208  ∃wex 1780

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-6 1970

This theorem depends on definitions:  df-bi 209  df-ex 1781

This theorem is referenced by:  elirrv  9060  bnj1014  32233  eusnsn  43310