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Mirrors > Home > ILE Home > Th. List > ax11e | GIF version |
Description: Analogue to ax-11 1397 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 1676 | . . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)) | |
2 | 1 | 19.21bi 1450 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 → ∃𝑦𝜑)) |
3 | 2 | com12 27 | 1 ⊢ (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑦𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∃wex 1381 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-11 1397 ax-4 1400 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: ax10oe 1678 drex1 1679 sbcof2 1691 ax11ev 1709 |
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