Theorem bj-alequex 37447

Description: A fol lemma. See alequexv 2030 for a version with a disjoint variable condition requiring fewer axioms. Can be used to reduce the proof of spimt 2417 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 2414	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		exim 1863	. 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 1567  ∃wex 1808

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-12 2212  ax-13 2403

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809

This theorem is used by:  bj-spimt2  37448  bj-equsal1t  37485