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Theorem ax10oe 1694
 Description: Quantifier Substitution for existential quantifiers. Analogue to ax10o 1619 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 105 . . . 4 (𝑥 = 𝑦 → (𝜓 → (𝑥 = 𝑦𝜓)))
21alimi 1360 . . 3 (∀𝑥 𝑥 = 𝑦 → ∀𝑥(𝜓 → (𝑥 = 𝑦𝜓)))
3 exim 1506 . . 3 (∀𝑥(𝜓 → (𝑥 = 𝑦𝜓)) → (∃𝑥𝜓 → ∃𝑥(𝑥 = 𝑦𝜓)))
42, 3syl 14 . 2 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜓 → ∃𝑥(𝑥 = 𝑦𝜓)))
5 ax11e 1693 . . 3 (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜓) → ∃𝑦𝜓))
65sps 1446 . 2 (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜓) → ∃𝑦𝜓))
74, 6syld 44 1 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜓 → ∃𝑦𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101  ∀wal 1257   = wceq 1259  ∃wex 1397 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-11 1413  ax-4 1416  ax-ial 1443 This theorem depends on definitions:  df-bi 114 This theorem is referenced by: (None)
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