Theorem alequexv 2007

Description: Version of equs4v 2006 with its consequence simplified by exsimpr 1870. (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 1972	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		exim 1834	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑))
3	1, 2	mpi 20	1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1535  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-6 1970

This theorem depends on definitions:  df-bi 209  df-ex 1781

This theorem is referenced by:  exsbim  2008  spsbe  2088  19.8a  2180  bj-spimvwt  34002