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Theorem equs4 2439
Description: Lemma used in proofs of implicit substitution properties. The converse requires either a disjoint variable condition (sb56 2278) or a non-freeness hypothesis (equs45f 2483). Usage of this theorem is discouraged because it depends on ax-13 2391. 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 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))

Proof of Theorem equs4
StepHypRef Expression
1 ax6e 2402 . 2 𝑥 𝑥 = 𝑦
2 exintr 1894 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥(𝑥 = 𝑦𝜑)))
31, 2mpi 20 1 (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff setvar class
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-5 1912  ax-6 1971  ax-7 2016  ax-12 2178  ax-13 2391
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782
This theorem is referenced by:  equsex  2441  equs45f  2483  equs5  2484  sb1OLD  2508  sb2ALT  2588  bj-sbsb  34169
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