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Mirrors > Home > ILE Home > Th. List > Mathboxes > strcoll2 | GIF version |
Description: Version of ax-strcoll 10107 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 2505 | . . 3 ⊢ (𝑧 = 𝑎 → (∀𝑥 ∈ 𝑧 ∃𝑦𝜑 ↔ ∀𝑥 ∈ 𝑎 ∃𝑦𝜑)) | |
2 | rexeq 2506 | . . . . . 6 ⊢ (𝑧 = 𝑎 → (∃𝑥 ∈ 𝑧 𝜑 ↔ ∃𝑥 ∈ 𝑎 𝜑)) | |
3 | 2 | bibi2d 221 | . . . . 5 ⊢ (𝑧 = 𝑎 → ((𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑧 𝜑) ↔ (𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑))) |
4 | 3 | albidv 1705 | . . . 4 ⊢ (𝑧 = 𝑎 → (∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑧 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑))) |
5 | 4 | exbidv 1706 | . . 3 ⊢ (𝑧 = 𝑎 → (∃𝑏∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑧 𝜑) ↔ ∃𝑏∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑))) |
6 | 1, 5 | imbi12d 223 | . 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 10107 | . . 3 ⊢ ∀𝑧(∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∃𝑏∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑧 𝜑)) | |
8 | 7 | spi 1429 | . 2 ⊢ (∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∃𝑏∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑧 𝜑)) |
9 | 6, 8 | chvarv 1812 | 1 ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∀wal 1241 ∃wex 1381 ∀wral 2306 ∃wrex 2307 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-strcoll 10107 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 |
This theorem is referenced by: strcollnft 10109 strcollnfALT 10111 |
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