Theorem bj-alequex 37274

Description: A fol lemma. See alequexv 2022 for a version with a disjoint variable condition requiring fewer axioms. Can be used to reduce the proof of spimt 2418 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 2415	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		exim 1855	. 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 1559  ∃wex 1800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-12 2213  ax-13 2404

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1801

This theorem is referenced by:  bj-spimt2  37275  bj-equsal1t  37312