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Mirrors > Home > ILE Home > Th. List > Mathboxes > strcollnf | GIF version |
Description: Version of ax-strcoll 12765 with one disjoint variable condition removed, the other disjoint variable condition replaced with 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 12767 | . 2 ⊢ (∀𝑥∀𝑦Ⅎ𝑏𝜑 → (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑))) | |
2 | strcollnf.nf | . . 3 ⊢ Ⅎ𝑏𝜑 | |
3 | 2 | ax-gen 1393 | . 2 ⊢ ∀𝑦Ⅎ𝑏𝜑 |
4 | 1, 3 | mpg 1395 | 1 ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1297 Ⅎwnf 1404 ∃wex 1436 ∀wral 2375 ∃wrex 2376 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-strcoll 12765 |
This theorem depends on definitions: df-bi 116 df-tru 1302 df-nf 1405 df-sb 1704 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 |
This theorem is referenced by: (None) |
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