Theorem bj-speiv 32724

Description: Version of spei 2261 with a dv condition, which does not require ax-13 2246 (neither ax-7 1935 nor ax-12 2047). (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
bj-speiv	⊢ ∃𝑥𝜑

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem bj-speiv

Step	Hyp	Ref	Expression
1		ax6ev 1890	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		bj-speiv.2	. . 3 ⊢ 𝜓
3		bj-speiv.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	2, 3	mpbiri 248	. 2 ⊢ (𝑥 = 𝑦 → 𝜑)
5	1, 4	eximii 1764	1 ⊢ ∃𝑥𝜑

Syntax hints:   → wi 4   ↔ wb 196  ∃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-6 1888

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by: (None)