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Mirrors > Home > ILE Home > Th. List > Mathboxes > strcollnf | GIF version |
Description: Version of ax-strcoll 10777 with one DV condition removed, the other DV condition replaced by a non-freeness hypothesis, and without initial universal quantifier. (Contributed by BJ, 21-Oct-2019.) |
Ref | Expression |
---|---|
strcollnf.nf | ⊢ Ⅎ𝑏𝜑 |
Ref | Expression |
---|---|
strcollnf | ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strcollnft 10779 | . 2 ⊢ (∀𝑥∀𝑦Ⅎ𝑏𝜑 → (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑))) | |
2 | strcollnf.nf | . . 3 ⊢ Ⅎ𝑏𝜑 | |
3 | 2 | ax-gen 1378 | . 2 ⊢ ∀𝑦Ⅎ𝑏𝜑 |
4 | 1, 3 | mpg 1380 | 1 ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∀wal 1282 Ⅎwnf 1389 ∃wex 1421 ∀wral 2348 ∃wrex 2349 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-strcoll 10777 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 |
This theorem is referenced by: (None) |
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