Theorem bj-alequex 37207

Description: A fol lemma. See alequexv 2011 for a version with a disjoint variable condition requiring fewer axioms. Can be used to reduce the proof of spimt 2407 from 133 to 112 bytes. (Contributed by BJ, 6-Oct-2018.)

Assertion

Ref	Expression
bj-alequex	⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑)

Proof of Theorem bj-alequex

Step	Hyp	Ref	Expression
1		ax6e 2404	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		exim 1844	. 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 1548  ∃wex 1789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-12 2202  ax-13 2393

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1790

This theorem is referenced by:  bj-spimt2  37208  bj-equsal1t  37245