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Mirrors > Home > ILE Home > Th. List > ax10oe | GIF version |
Description: Quantifier Substitution for existential quantifiers. Analogue to ax10o 1603 but for ∃ rather than ∀. (Contributed by Jim Kingdon, 21-Dec-2017.) |
Ref | Expression |
---|---|
ax10oe | ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜓 → ∃𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-ia3 101 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 → (𝑥 = 𝑦 ∧ 𝜓))) | |
2 | 1 | alimi 1344 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑥(𝜓 → (𝑥 = 𝑦 ∧ 𝜓))) |
3 | exim 1490 | . . 3 ⊢ (∀𝑥(𝜓 → (𝑥 = 𝑦 ∧ 𝜓)) → (∃𝑥𝜓 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))) | |
4 | 2, 3 | syl 14 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜓 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))) |
5 | ax11e 1677 | . . 3 ⊢ (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) → ∃𝑦𝜓)) | |
6 | 5 | sps 1430 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) → ∃𝑦𝜓)) |
7 | 4, 6 | syld 40 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜓 → ∃𝑦𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∀wal 1241 = 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: (None) |
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