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

Step	Hyp	Ref	Expression
1		ax-ia3 101	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 → (𝑥 = 𝑦 ∧ 𝜓)))
2	1	alimi 1344	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑥(𝜓 → (𝑥 = 𝑦 ∧ 𝜓)))
3		exim 1490	. . 3 ⊢ (∀𝑥(𝜓 → (𝑥 = 𝑦 ∧ 𝜓)) → (∃𝑥𝜓 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)))
4	2, 3	syl 14	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜓 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)))
5		ax11e 1677	. . 3 ⊢ (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) → ∃𝑦𝜓))
6	5	sps 1430	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) → ∃𝑦𝜓))
7	4, 6	syld 40	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜓 → ∃𝑦𝜓))

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)