Theorem ax10oe 1811

Description: Quantifier Substitution for existential quantifiers. Analogue to ax10o 1729 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 108	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 → (𝑥 = 𝑦 ∧ 𝜓)))
2	1	alimi 1469	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑥(𝜓 → (𝑥 = 𝑦 ∧ 𝜓)))
3		exim 1613	. . 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 1810	. . 3 ⊢ (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) → ∃𝑦𝜓))
6	5	sps 1551	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) → ∃𝑦𝜓))
7	4, 6	syld 45	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜓 → ∃𝑦𝜓))

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1362   = wceq 1364  ∃wex 1506

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-11 1520  ax-4 1524  ax-ial 1548

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)