Theorem speivw 1972

Description: Version of spei 2398 with a disjoint variable condition, which does not require ax-13 2376 (neither ax-7 2006 nor ax-12 2176). (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 248	. 2 ⊢ (𝑥 = 𝑦 → (𝜓 → 𝜑))
3		speivw.2	. 2 ⊢ 𝜓
4	2, 3	speiv 1971	1 ⊢ ∃𝑥𝜑

Syntax hints:   → wi 4   ↔ wb 206  ∃wex 1778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-6 1966

This theorem depends on definitions:  df-bi 207  df-ex 1779

This theorem is referenced by:  elirrv  9637  bnj1014  34976  eusnsn  47043