Theorem spei 2408

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 2386. Use the weaker speiv 1972 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 2397	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		spei.2	. . 3 ⊢ 𝜓
3		spei.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	2, 3	mpbiri 260	. 2 ⊢ (𝑥 = 𝑦 → 𝜑)
5	1, 4	eximii 1833	1 ⊢ ∃𝑥𝜑

Syntax hints:   → wi 4   ↔ wb 208  ∃wex 1776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-12 2172  ax-13 2386

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777

This theorem is referenced by: (None)