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Theorem equs4 1725
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 1696 . . 3 𝑥 𝑥 = 𝑦
2 19.29 1620 . . 3 ((∀𝑥(𝑥 = 𝑦𝜑) ∧ ∃𝑥 𝑥 = 𝑦) → ∃𝑥((𝑥 = 𝑦𝜑) ∧ 𝑥 = 𝑦))
31, 2mpan2 425 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥((𝑥 = 𝑦𝜑) ∧ 𝑥 = 𝑦))
4 ancl 318 . . . 4 ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦 → (𝑥 = 𝑦𝜑)))
54imp 124 . . 3 (((𝑥 = 𝑦𝜑) ∧ 𝑥 = 𝑦) → (𝑥 = 𝑦𝜑))
65eximi 1600 . 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 104  wal 1351   = wceq 1353  wex 1492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-i9 1530  ax-ial 1534
This theorem depends on definitions:  df-bi 117
This theorem is referenced by:  sb2  1767  equs45f  1802
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