Theorem bj-alequexv 32655

Description: Version of bj-alequex 32708 with DV(x,y), requiring fewer axioms. (Contributed by BJ, 9-Nov-2021.)

Assertion

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

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-alequexv

Step	Hyp	Ref	Expression
1		ax6ev 1890	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		exim 1761	. 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 1481  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-6 1888

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by:  bj-spimvwt  32656