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Theorem ax10oe 1807
Description: Quantifier Substitution for existential quantifiers. Analogue to ax10o 1725 but for rather than . (Contributed by Jim Kingdon, 21-Dec-2017.)
Assertion
Ref Expression
ax10oe (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜓 → ∃𝑦𝜓))

Proof of Theorem ax10oe
StepHypRef Expression
1 ax-ia3 108 . . . 4 (𝑥 = 𝑦 → (𝜓 → (𝑥 = 𝑦𝜓)))
21alimi 1465 . . 3 (∀𝑥 𝑥 = 𝑦 → ∀𝑥(𝜓 → (𝑥 = 𝑦𝜓)))
3 exim 1609 . . 3 (∀𝑥(𝜓 → (𝑥 = 𝑦𝜓)) → (∃𝑥𝜓 → ∃𝑥(𝑥 = 𝑦𝜓)))
42, 3syl 14 . 2 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜓 → ∃𝑥(𝑥 = 𝑦𝜓)))
5 ax11e 1806 . . 3 (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜓) → ∃𝑦𝜓))
65sps 1547 . 2 (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜓) → ∃𝑦𝜓))
74, 6syld 45 1 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜓 → ∃𝑦𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wal 1361   = wceq 1363  wex 1502
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 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-11 1516  ax-4 1520  ax-ial 1544
This theorem depends on definitions:  df-bi 117
This theorem is referenced by: (None)
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