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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj985v | Structured version Visualization version GIF version |
Description: Version of bnj985 32505 with an additional disjoint variable condition, not requiring ax-13 2372. (Contributed by Gino Giotto, 27-Mar-2024.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj985v.3 | ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
bnj985v.6 | ⊢ (𝜒′ ↔ [𝑝 / 𝑛]𝜒) |
bnj985v.9 | ⊢ (𝜒″ ↔ [𝐺 / 𝑓]𝜒′) |
bnj985v.11 | ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} |
bnj985v.13 | ⊢ 𝐺 = (𝑓 ∪ {〈𝑛, 𝐶〉}) |
Ref | Expression |
---|---|
bnj985v | ⊢ (𝐺 ∈ 𝐵 ↔ ∃𝑝𝜒″) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj985v.13 | . . . 4 ⊢ 𝐺 = (𝑓 ∪ {〈𝑛, 𝐶〉}) | |
2 | 1 | bnj918 32316 | . . 3 ⊢ 𝐺 ∈ V |
3 | bnj985v.3 | . . . 4 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) | |
4 | bnj985v.11 | . . . 4 ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} | |
5 | 3, 4 | bnj984 32503 | . . 3 ⊢ (𝐺 ∈ V → (𝐺 ∈ 𝐵 ↔ [𝐺 / 𝑓]∃𝑛𝜒)) |
6 | 2, 5 | ax-mp 5 | . 2 ⊢ (𝐺 ∈ 𝐵 ↔ [𝐺 / 𝑓]∃𝑛𝜒) |
7 | sbcex2 3743 | . . 3 ⊢ ([𝐺 / 𝑓]∃𝑝𝜒′ ↔ ∃𝑝[𝐺 / 𝑓]𝜒′) | |
8 | nfv 1921 | . . . . . . 7 ⊢ Ⅎ𝑝𝜒 | |
9 | 8 | sb8ev 2356 | . . . . . 6 ⊢ (∃𝑛𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒) |
10 | sbsbc 3684 | . . . . . . 7 ⊢ ([𝑝 / 𝑛]𝜒 ↔ [𝑝 / 𝑛]𝜒) | |
11 | 10 | exbii 1854 | . . . . . 6 ⊢ (∃𝑝[𝑝 / 𝑛]𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒) |
12 | 9, 11 | bitri 278 | . . . . 5 ⊢ (∃𝑛𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒) |
13 | bnj985v.6 | . . . . 5 ⊢ (𝜒′ ↔ [𝑝 / 𝑛]𝜒) | |
14 | 12, 13 | bnj133 32276 | . . . 4 ⊢ (∃𝑛𝜒 ↔ ∃𝑝𝜒′) |
15 | 14 | sbcbii 3738 | . . 3 ⊢ ([𝐺 / 𝑓]∃𝑛𝜒 ↔ [𝐺 / 𝑓]∃𝑝𝜒′) |
16 | bnj985v.9 | . . . 4 ⊢ (𝜒″ ↔ [𝐺 / 𝑓]𝜒′) | |
17 | 16 | exbii 1854 | . . 3 ⊢ (∃𝑝𝜒″ ↔ ∃𝑝[𝐺 / 𝑓]𝜒′) |
18 | 7, 15, 17 | 3bitr4i 306 | . 2 ⊢ ([𝐺 / 𝑓]∃𝑛𝜒 ↔ ∃𝑝𝜒″) |
19 | 6, 18 | bitri 278 | 1 ⊢ (𝐺 ∈ 𝐵 ↔ ∃𝑝𝜒″) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ w3a 1088 = wceq 1542 ∃wex 1786 [wsb 2074 ∈ wcel 2114 {cab 2716 ∃wrex 3054 Vcvv 3398 [wsbc 3680 ∪ cun 3841 {csn 4516 〈cop 4522 Fn wfn 6334 ∧ w-bnj17 32235 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-rex 3059 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-sn 4517 df-pr 4519 df-uni 4797 df-bnj17 32236 |
This theorem is referenced by: bnj1018 32515 |
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