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Mirrors > Home > ILE Home > Th. List > Mathboxes > strcollnf | GIF version |
Description: Version of ax-strcoll 15030 with one disjoint variable condition
removed,
the other disjoint variable condition replaced with a nonfreeness
hypothesis, and without initial universal quantifier. Version of
strcoll2 15031 with the disjoint variable condition on
𝑏, 𝜑 replaced
with a nonfreeness hypothesis.
This proof aims to demonstrate a standard technique, but strcoll2 15031 will generally suffice: since the theorem asserts the existence of a set 𝑏, supposing that that setvar does not occur in the already defined 𝜑 is not a big constraint. (Contributed by BJ, 21-Oct-2019.) |
Ref | Expression |
---|---|
strcollnf.nf | ⊢ Ⅎ𝑏𝜑 |
Ref | Expression |
---|---|
strcollnf | ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strcollnft 15032 | . 2 ⊢ (∀𝑥∀𝑦Ⅎ𝑏𝜑 → (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑))) | |
2 | strcollnf.nf | . . 3 ⊢ Ⅎ𝑏𝜑 | |
3 | 2 | ax-gen 1459 | . 2 ⊢ ∀𝑦Ⅎ𝑏𝜑 |
4 | 1, 3 | mpg 1461 | 1 ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∀wal 1361 Ⅎwnf 1470 ∃wex 1502 ∀wral 2465 ∃wrex 2466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 ax-strcoll 15030 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 |
This theorem is referenced by: (None) |
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