Theorem spei 2428

Description: Inference from existential specialization, using implicit substitution. Remove a distinct variable constraint. Usage of this theorem is discouraged because it depends on ax-13 2406. Use the weaker speiv 2005 if possible. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
spei	⊢ ∃𝑥𝜑

Proof of Theorem spei

Step	Hyp	Ref	Expression
1		ax6e 2417	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		spei.2	. . 3 ⊢ 𝜓
3		spei.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	2, 3	mpbiri 261	. 2 ⊢ (𝑥 = 𝑦 → 𝜑)
5	1, 4	eximii 1870	1 ⊢ ∃𝑥𝜑

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∃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-5 1943  ax-6 2000  ax-7 2041  ax-12 2216  ax-13 2406

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by: (None)