Theorem ax11e 1789

Description: Analogue to ax-11 1499 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 1788	. . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
2	1	19.21bi 1551	. 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 103   = wceq 1348  ∃wex 1485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-11 1499  ax-4 1503  ax-ial 1527

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  ax10oe  1790  drex1  1791  sbcof2  1803  ax11ev  1821