Theorem ax11e 1807

Description: Analogue to ax-11 1517 but for existential quantification. (Contributed by Mario Carneiro and Jim Kingdon, 31-Dec-2017.) (Proved by Mario Carneiro, 9-Feb-2018.)

Assertion

Ref	Expression
ax11e	⊢ (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑦𝜑))

Proof of Theorem ax11e

Step	Hyp	Ref	Expression
1		equs5e 1806	. . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
2	1	19.21bi 1569	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 → ∃𝑦𝜑))
3	2	com12 30	1 ⊢ (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑦𝜑))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364  ∃wex 1503

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-11 1517  ax-4 1521  ax-ial 1545

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  ax10oe  1808  drex1  1809  sbcof2  1821  ax11ev  1839