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Mirrors > Home > ILE Home > Th. List > ax11e | GIF version |
Description: Analogue to ax-11 1486 but for existential quantification. (Contributed by Mario Carneiro and Jim Kingdon, 31-Dec-2017.) (Proved by Mario Carneiro, 9-Feb-2018.) |
Ref | Expression |
---|---|
ax11e | ⊢ (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑦𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equs5e 1775 | . . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)) | |
2 | 1 | 19.21bi 1538 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 → ∃𝑦𝜑)) |
3 | 2 | com12 30 | 1 ⊢ (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑦𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1335 ∃wex 1472 |
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 1427 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-11 1486 ax-4 1490 ax-ial 1514 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: ax10oe 1777 drex1 1778 sbcof2 1790 ax11ev 1808 |
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