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Theorem ax10oe 1777
 Description: Quantifier Substitution for existential quantifiers. Analogue to ax10o 1695 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 107 . . . 4 (𝑥 = 𝑦 → (𝜓 → (𝑥 = 𝑦𝜓)))
21alimi 1435 . . 3 (∀𝑥 𝑥 = 𝑦 → ∀𝑥(𝜓 → (𝑥 = 𝑦𝜓)))
3 exim 1579 . . 3 (∀𝑥(𝜓 → (𝑥 = 𝑦𝜓)) → (∃𝑥𝜓 → ∃𝑥(𝑥 = 𝑦𝜓)))
42, 3syl 14 . 2 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜓 → ∃𝑥(𝑥 = 𝑦𝜓)))
5 ax11e 1776 . . 3 (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜓) → ∃𝑦𝜓))
65sps 1517 . 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 103  ∀wal 1333   = wceq 1335  ∃wex 1472 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-11 1486  ax-4 1490  ax-ial 1514 This theorem depends on definitions:  df-bi 116 This theorem is referenced by: (None)
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