Theorem alequexv 2034

Description: Version of equs4v 2033 with its consequence simplified by exsimpr 1902. (Contributed by BJ, 9-Nov-2021.)

Assertion

Ref	Expression
alequexv	⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)

Proof of Theorem alequexv

Step	Hyp	Ref	Expression
1		ax6ev 2002	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		exim 1867	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑))
3	1, 2	mpi 21	1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃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:  exsbim  2035  spsbe  2119  19.8a  2220  bj-spimvwt  37351