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Theorem equs4 2416
Description: Lemma used in proofs of implicit substitution properties. The converse requires either a disjoint variable condition (sbalex 2244) or a nonfreeness hypothesis (equs45f 2459). Usage of this theorem is discouraged because it depends on ax-13 2372. See equs4v 2011 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 2383 . 2 𝑥 𝑥 = 𝑦
2 exintr 1899 . 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 1540  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 1975  ax-7 2020  ax-12 2179  ax-13 2372
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1787
This theorem is referenced by:  equsex  2418  equs45f  2459  equs5  2460  sb1OLD  2482  bj-sbsb  34651
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