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Theorem equs4 2380
 Description: Lemma used in proofs of implicit substitution properties. The converse requires either a disjoint variable condition (sb56 2247) or a non-freeness hypothesis (equs45f 2424). See equs4v 2049 for a 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.)
Assertion
Ref Expression
equs4 (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))

Proof of Theorem equs4
StepHypRef Expression
1 ax6e 2346 . 2 𝑥 𝑥 = 𝑦
2 exintr 1938 . 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 386  ∀wal 1599  ∃wex 1823 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-12 2162  ax-13 2333 This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824 This theorem is referenced by:  equsex  2382  equs45f  2424  equs5  2425  sb2  2426  bj-sbsb  33413
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