Theorem equs4 2447

Description: Lemma used in proofs of implicit substitution properties. The converse requires either a disjoint variable condition (sbalex 2277) or a nonfreeness hypothesis (equs45f 2490). Usage of this theorem is discouraged because it depends on ax-13 2403. See equs4v 2020 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 2414	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		exintr 1912	. 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 1558  ∃wex 1799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-12 2212  ax-13 2403

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1800

This theorem is referenced by:  equsex  2449  equs45f  2490  equs5  2491  bj-sbsb  37322