Theorem equs4v 2029

Description: Version of equs4 2447 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 1998	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		exintr 1921	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
3	1, 2	mpi 21	1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400  ∀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-6 1996

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

This theorem is used by:  sb1v  2120  sbalex  2277  sbalexOLD  2278  equsexv  2303  bj-subst  37311  bj-equs45fv  37474