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Theorem ax11ev 1725
Description: Analogue to ax11v 1724 for existential quantification. (Contributed by Jim Kingdon, 9-Jan-2018.)
Assertion
Ref Expression
ax11ev (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → 𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem ax11ev
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 a9e 1602 . 2 𝑧 𝑧 = 𝑦
2 ax11e 1693 . . . . 5 (𝑥 = 𝑧 → (∃𝑥(𝑥 = 𝑧𝜑) → ∃𝑧𝜑))
3 ax-17 1435 . . . . . 6 (𝜑 → ∀𝑧𝜑)
4319.9h 1550 . . . . 5 (∃𝑧𝜑𝜑)
52, 4syl6ib 154 . . . 4 (𝑥 = 𝑧 → (∃𝑥(𝑥 = 𝑧𝜑) → 𝜑))
6 equequ2 1615 . . . . 5 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
76anbi1d 446 . . . . . . 7 (𝑧 = 𝑦 → ((𝑥 = 𝑧𝜑) ↔ (𝑥 = 𝑦𝜑)))
87exbidv 1722 . . . . . 6 (𝑧 = 𝑦 → (∃𝑥(𝑥 = 𝑧𝜑) ↔ ∃𝑥(𝑥 = 𝑦𝜑)))
98imbi1d 224 . . . . 5 (𝑧 = 𝑦 → ((∃𝑥(𝑥 = 𝑧𝜑) → 𝜑) ↔ (∃𝑥(𝑥 = 𝑦𝜑) → 𝜑)))
106, 9imbi12d 227 . . . 4 (𝑧 = 𝑦 → ((𝑥 = 𝑧 → (∃𝑥(𝑥 = 𝑧𝜑) → 𝜑)) ↔ (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → 𝜑))))
115, 10mpbii 140 . . 3 (𝑧 = 𝑦 → (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → 𝜑)))
1211exlimiv 1505 . 2 (∃𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → 𝜑)))
131, 12ax-mp 7 1 (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wex 1397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443
This theorem depends on definitions:  df-bi 114
This theorem is referenced by: (None)
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