Theorem speiv 2005

Description: Inference from existential specialization. (Contributed by NM, 19-Aug-1993.) Use spimew 2004. (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 1836	. . 3 ⊢ (𝜓 → ∀𝑥𝜓)
3		speiv.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜓 → 𝜑))
4	2, 3	spimew 2004	. 2 ⊢ (𝜓 → ∃𝑥𝜑)
5	1, 4	ax-mp 5	1 ⊢ ∃𝑥𝜑

This proof depends on syntax axioms:   → wi 4  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-6 2000

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  speivw  2006  exgen  2007