Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > Mathboxes > strcoll2 | GIF version |
Description: Version of ax-strcoll 10777 with one DV condition removed and without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.) |
Ref | Expression |
---|---|
strcoll2 | ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | raleq 2549 | . . 3 ⊢ (𝑧 = 𝑎 → (∀𝑥 ∈ 𝑧 ∃𝑦𝜑 ↔ ∀𝑥 ∈ 𝑎 ∃𝑦𝜑)) | |
2 | rexeq 2550 | . . . . . 6 ⊢ (𝑧 = 𝑎 → (∃𝑥 ∈ 𝑧 𝜑 ↔ ∃𝑥 ∈ 𝑎 𝜑)) | |
3 | 2 | bibi2d 230 | . . . . 5 ⊢ (𝑧 = 𝑎 → ((𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑧 𝜑) ↔ (𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑))) |
4 | 3 | albidv 1745 | . . . 4 ⊢ (𝑧 = 𝑎 → (∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑧 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑))) |
5 | 4 | exbidv 1746 | . . 3 ⊢ (𝑧 = 𝑎 → (∃𝑏∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑧 𝜑) ↔ ∃𝑏∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑))) |
6 | 1, 5 | imbi12d 232 | . 2 ⊢ (𝑧 = 𝑎 → ((∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∃𝑏∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑧 𝜑)) ↔ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑)))) |
7 | ax-strcoll 10777 | . . 3 ⊢ ∀𝑧(∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∃𝑏∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑧 𝜑)) | |
8 | 7 | spi 1469 | . 2 ⊢ (∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∃𝑏∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑧 𝜑)) |
9 | 6, 8 | chvarv 1853 | 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 ∃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-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: strcollnft 10779 strcollnfALT 10781 |
Copyright terms: Public domain | W3C validator |