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Theorem equs5e 1723
Description: A property related to substitution that unlike equs5 1757 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) (Revised by NM, 3-Feb-2015.)
Assertion
Ref Expression
equs5e (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))

Proof of Theorem equs5e
StepHypRef Expression
1 19.8a 1527 . . . . 5 (𝜑 → ∃𝑦𝜑)
2 hbe1 1429 . . . . 5 (∃𝑦𝜑 → ∀𝑦𝑦𝜑)
31, 2syl 14 . . . 4 (𝜑 → ∀𝑦𝑦𝜑)
43anim2i 334 . . 3 ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦 ∧ ∀𝑦𝑦𝜑))
54eximi 1536 . 2 (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝑦𝜑))
6 equs5a 1722 . 2 (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
75, 6syl 14 1 (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1287   = wceq 1289  wex 1426
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 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-11 1442  ax-4 1445  ax-ial 1472
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  ax11e  1724  sb4e  1733
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