Theorem speiv 1977

Description: Inference from existential specialization. (Contributed by NM, 19-Aug-1993.) (Revised by Wolf Lammen, 22-Oct-2023.)

Hypotheses

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

Assertion

Ref	Expression
speiv	⊢ ∃𝑥𝜑

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem speiv

Step	Hyp	Ref	Expression
1		speiv.2	. 2 ⊢ 𝜓
2	1	hbth 1807	. . 3 ⊢ (𝜓 → ∀𝑥𝜓)
3		speiv.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜓 → 𝜑))
4	2, 3	spimew 1976	. 2 ⊢ (𝜓 → ∃𝑥𝜑)
5	1, 4	ax-mp 5	1 ⊢ ∃𝑥𝜑

Syntax hints:   → wi 4  ∃wex 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-6 1972

This theorem depends on definitions:  df-bi 206  df-ex 1784

This theorem is referenced by:  speivw  1978  exgen  1979