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Mirrors > Home > MPE Home > Th. List > Mathboxes > 19.9dev | Structured version Visualization version GIF version |
Description: 19.9d 2202 in the case of an existential quantifier, avoiding the ax-10 2143 from nfex 2333 that would be used for the hypothesis of 19.9d 2202, at the cost of an additional DV condition on 𝑦, 𝜑. (Contributed by SN, 26-May-2024.) |
Ref | Expression |
---|---|
19.9dev.1 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
19.9dev | ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 ↔ ∃𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | excom 2167 | . 2 ⊢ (∃𝑥∃𝑦𝜓 ↔ ∃𝑦∃𝑥𝜓) | |
2 | 19.9dev.1 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
3 | 19.9t 2203 | . . . 4 ⊢ (Ⅎ𝑥𝜓 → (∃𝑥𝜓 ↔ 𝜓)) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝜑 → (∃𝑥𝜓 ↔ 𝜓)) |
5 | 4 | exbidv 1923 | . 2 ⊢ (𝜑 → (∃𝑦∃𝑥𝜓 ↔ ∃𝑦𝜓)) |
6 | 1, 5 | syl5bb 286 | 1 ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 ↔ ∃𝑦𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∃wex 1782 Ⅎwnf 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-11 2159 ax-12 2176 |
This theorem depends on definitions: df-bi 210 df-ex 1783 df-nf 1787 |
This theorem is referenced by: (None) |
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