Theorem equs4 2424

Description: Lemma used in proofs of implicit substitution properties. The converse requires either a disjoint variable condition (sbalex 2254) or a nonfreeness hypothesis (equs45f 2467). Usage of this theorem is discouraged because it depends on ax-13 2380. See equs4v 2007 for a weaker version requiring fewer axioms. (Contributed by NM, 10-May-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.) (Proof shortened by Wolf Lammen, 5-Feb-2018.) (New usage is discouraged.)

Assertion

Ref	Expression
equs4	⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))

Proof of Theorem equs4

Step	Hyp	Ref	Expression
1		ax6e 2391	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		exintr 1899	. 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 396  ∀wal 1545  ∃wex 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-12 2189  ax-13 2380

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787

This theorem is referenced by:  equsex  2426  equs45f  2467  equs5  2468  bj-sbsb  37190