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Mirrors > Home > ILE Home > Th. List > Mathboxes > strcoll2 | GIF version |
Description: Version of ax-strcoll 13568 with one disjoint variable condition removed and without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.) |
Ref | Expression |
---|---|
strcoll2 | ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | raleq 2652 | . . 3 ⊢ (𝑧 = 𝑎 → (∀𝑥 ∈ 𝑧 ∃𝑦𝜑 ↔ ∀𝑥 ∈ 𝑎 ∃𝑦𝜑)) | |
2 | raleq 2652 | . . . . 5 ⊢ (𝑧 = 𝑎 → (∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑏 𝜑 ↔ ∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑)) | |
3 | rexeq 2653 | . . . . . 6 ⊢ (𝑧 = 𝑎 → (∃𝑥 ∈ 𝑧 𝜑 ↔ ∃𝑥 ∈ 𝑎 𝜑)) | |
4 | 3 | ralbidv 2457 | . . . . 5 ⊢ (𝑧 = 𝑎 → (∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑧 𝜑 ↔ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑)) |
5 | 2, 4 | anbi12d 465 | . . . 4 ⊢ (𝑧 = 𝑎 → ((∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑧 𝜑) ↔ (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑))) |
6 | 5 | exbidv 1805 | . . 3 ⊢ (𝑧 = 𝑎 → (∃𝑏(∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑧 𝜑) ↔ ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑))) |
7 | 1, 6 | imbi12d 233 | . 2 ⊢ (𝑧 = 𝑎 → ((∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑧 𝜑)) ↔ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑)))) |
8 | ax-strcoll 13568 | . . 3 ⊢ ∀𝑧(∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑧 𝜑)) | |
9 | 8 | spi 1516 | . 2 ⊢ (∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑧 𝜑)) |
10 | 7, 9 | chvarv 1917 | 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 103 ∃wex 1472 ∀wral 2435 ∃wrex 2436 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 ax-strcoll 13568 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 |
This theorem is referenced by: strcollnft 13570 strcollnfALT 13572 |
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