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Theorem equs4 1655
 Description: Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.)
Assertion
Ref Expression
equs4 (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))

Proof of Theorem equs4
StepHypRef Expression
1 a9e 1627 . . 3 𝑥 𝑥 = 𝑦
2 19.29 1552 . . 3 ((∀𝑥(𝑥 = 𝑦𝜑) ∧ ∃𝑥 𝑥 = 𝑦) → ∃𝑥((𝑥 = 𝑦𝜑) ∧ 𝑥 = 𝑦))
31, 2mpan2 416 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥((𝑥 = 𝑦𝜑) ∧ 𝑥 = 𝑦))
4 ancl 311 . . . 4 ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦 → (𝑥 = 𝑦𝜑)))
54imp 122 . . 3 (((𝑥 = 𝑦𝜑) ∧ 𝑥 = 𝑦) → (𝑥 = 𝑦𝜑))
65eximi 1532 . 2 (∃𝑥((𝑥 = 𝑦𝜑) ∧ 𝑥 = 𝑦) → ∃𝑥(𝑥 = 𝑦𝜑))
73, 6syl 14 1 (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102  ∀wal 1283   = wceq 1285  ∃wex 1422 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-i9 1464  ax-ial 1468 This theorem depends on definitions:  df-bi 115 This theorem is referenced by:  sb2  1692  equs45f  1725
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