Theorem bj-alequex

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

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

Proof

Step	Hyp	Ref	Expression
1		ax6e	⊢ ∃ 𝑥 𝑥 = 𝑦
2		exim	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ( ∃ 𝑥 𝑥 = 𝑦 → ∃ 𝑥 𝜑 ) )
3	1 2	mpi	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ∃ 𝑥 𝜑 )