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Theorem ax11ev 1828
Description: Analogue to ax11v 1827 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 1696 . 2 𝑧 𝑧 = 𝑦
2 ax11e 1796 . . . . 5 (𝑥 = 𝑧 → (∃𝑥(𝑥 = 𝑧𝜑) → ∃𝑧𝜑))
3 ax-17 1526 . . . . . 6 (𝜑 → ∀𝑧𝜑)
4319.9h 1643 . . . . 5 (∃𝑧𝜑𝜑)
52, 4imbitrdi 161 . . . 4 (𝑥 = 𝑧 → (∃𝑥(𝑥 = 𝑧𝜑) → 𝜑))
6 equequ2 1713 . . . . 5 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
76anbi1d 465 . . . . . . 7 (𝑧 = 𝑦 → ((𝑥 = 𝑧𝜑) ↔ (𝑥 = 𝑦𝜑)))
87exbidv 1825 . . . . . 6 (𝑧 = 𝑦 → (∃𝑥(𝑥 = 𝑧𝜑) ↔ ∃𝑥(𝑥 = 𝑦𝜑)))
98imbi1d 231 . . . . 5 (𝑧 = 𝑦 → ((∃𝑥(𝑥 = 𝑧𝜑) → 𝜑) ↔ (∃𝑥(𝑥 = 𝑦𝜑) → 𝜑)))
106, 9imbi12d 234 . . . 4 (𝑧 = 𝑦 → ((𝑥 = 𝑧 → (∃𝑥(𝑥 = 𝑧𝜑) → 𝜑)) ↔ (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → 𝜑))))
115, 10mpbii 148 . . 3 (𝑧 = 𝑦 → (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → 𝜑)))
1211exlimiv 1598 . 2 (∃𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → 𝜑)))
131, 12ax-mp 5 1 (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wex 1492
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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534
This theorem depends on definitions:  df-bi 117
This theorem is referenced by: (None)
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