Theorem equs4v 2006

Description: Version of equs4 2427 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 10-May-1993.) (Revised by BJ, 31-May-2019.)

Assertion

Ref	Expression
equs4v	⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem equs4v

Step	Hyp	Ref	Expression
1		ax6ev 1972	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		exintr 1893	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
3	1, 2	mpi 20	1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-6 1970

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782

This theorem is referenced by:  sb1v  2092  sb2vOLDALT  2559  bj-sb56  34107  bj-equs45fv  34248