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Theorem ax11ev 1801
Description: Analogue to ax11v 1800 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 1675 . 2 𝑧 𝑧 = 𝑦
2 ax11e 1769 . . . . 5 (𝑥 = 𝑧 → (∃𝑥(𝑥 = 𝑧𝜑) → ∃𝑧𝜑))
3 ax-17 1507 . . . . . 6 (𝜑 → ∀𝑧𝜑)
4319.9h 1623 . . . . 5 (∃𝑧𝜑𝜑)
52, 4syl6ib 160 . . . 4 (𝑥 = 𝑧 → (∃𝑥(𝑥 = 𝑧𝜑) → 𝜑))
6 equequ2 1690 . . . . 5 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
76anbi1d 461 . . . . . . 7 (𝑧 = 𝑦 → ((𝑥 = 𝑧𝜑) ↔ (𝑥 = 𝑦𝜑)))
87exbidv 1798 . . . . . 6 (𝑧 = 𝑦 → (∃𝑥(𝑥 = 𝑧𝜑) ↔ ∃𝑥(𝑥 = 𝑦𝜑)))
98imbi1d 230 . . . . 5 (𝑧 = 𝑦 → ((∃𝑥(𝑥 = 𝑧𝜑) → 𝜑) ↔ (∃𝑥(𝑥 = 𝑦𝜑) → 𝜑)))
106, 9imbi12d 233 . . . 4 (𝑧 = 𝑦 → ((𝑥 = 𝑧 → (∃𝑥(𝑥 = 𝑧𝜑) → 𝜑)) ↔ (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → 𝜑))))
115, 10mpbii 147 . . 3 (𝑧 = 𝑦 → (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → 𝜑)))
1211exlimiv 1578 . 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 103   = wceq 1332  wex 1469
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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515
This theorem depends on definitions:  df-bi 116
This theorem is referenced by: (None)
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