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Theorem ax10oe 1678
 Description: Quantifier Substitution for existential quantifiers. Analogue to ax10o 1603 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 101 . . . 4 (𝑥 = 𝑦 → (𝜓 → (𝑥 = 𝑦𝜓)))
21alimi 1344 . . 3 (∀𝑥 𝑥 = 𝑦 → ∀𝑥(𝜓 → (𝑥 = 𝑦𝜓)))
3 exim 1490 . . 3 (∀𝑥(𝜓 → (𝑥 = 𝑦𝜓)) → (∃𝑥𝜓 → ∃𝑥(𝑥 = 𝑦𝜓)))
42, 3syl 14 . 2 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜓 → ∃𝑥(𝑥 = 𝑦𝜓)))
5 ax11e 1677 . . 3 (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜓) → ∃𝑦𝜓))
65sps 1430 . 2 (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜓) → ∃𝑦𝜓))
74, 6syld 40 1 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜓 → ∃𝑦𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1241   = wceq 1243  ∃wex 1381 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-11 1397  ax-4 1400  ax-ial 1427 This theorem depends on definitions:  df-bi 110 This theorem is referenced by: (None)
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