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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj985 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 32179. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj985.3 | ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
bnj985.6 | ⊢ (𝜒′ ↔ [𝑝 / 𝑛]𝜒) |
bnj985.9 | ⊢ (𝜒″ ↔ [𝐺 / 𝑓]𝜒′) |
bnj985.11 | ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} |
bnj985.13 | ⊢ 𝐺 = (𝑓 ∪ {〈𝑛, 𝐶〉}) |
Ref | Expression |
---|---|
bnj985 | ⊢ (𝐺 ∈ 𝐵 ↔ ∃𝑝𝜒″) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj985.13 | . . . 4 ⊢ 𝐺 = (𝑓 ∪ {〈𝑛, 𝐶〉}) | |
2 | 1 | bnj918 31936 | . . 3 ⊢ 𝐺 ∈ V |
3 | bnj985.3 | . . . 4 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) | |
4 | bnj985.11 | . . . 4 ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} | |
5 | 3, 4 | bnj984 32123 | . . 3 ⊢ (𝐺 ∈ V → (𝐺 ∈ 𝐵 ↔ [𝐺 / 𝑓]∃𝑛𝜒)) |
6 | 2, 5 | ax-mp 5 | . 2 ⊢ (𝐺 ∈ 𝐵 ↔ [𝐺 / 𝑓]∃𝑛𝜒) |
7 | sbcex2 3831 | . . 3 ⊢ ([𝐺 / 𝑓]∃𝑝𝜒′ ↔ ∃𝑝[𝐺 / 𝑓]𝜒′) | |
8 | nfv 1906 | . . . . . . 7 ⊢ Ⅎ𝑝𝜒 | |
9 | 8 | sb8e 2553 | . . . . . 6 ⊢ (∃𝑛𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒) |
10 | sbsbc 3773 | . . . . . . 7 ⊢ ([𝑝 / 𝑛]𝜒 ↔ [𝑝 / 𝑛]𝜒) | |
11 | 10 | exbii 1839 | . . . . . 6 ⊢ (∃𝑝[𝑝 / 𝑛]𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒) |
12 | 9, 11 | bitri 276 | . . . . 5 ⊢ (∃𝑛𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒) |
13 | bnj985.6 | . . . . 5 ⊢ (𝜒′ ↔ [𝑝 / 𝑛]𝜒) | |
14 | 12, 13 | bnj133 31896 | . . . 4 ⊢ (∃𝑛𝜒 ↔ ∃𝑝𝜒′) |
15 | 14 | sbcbii 3826 | . . 3 ⊢ ([𝐺 / 𝑓]∃𝑛𝜒 ↔ [𝐺 / 𝑓]∃𝑝𝜒′) |
16 | bnj985.9 | . . . 4 ⊢ (𝜒″ ↔ [𝐺 / 𝑓]𝜒′) | |
17 | 16 | exbii 1839 | . . 3 ⊢ (∃𝑝𝜒″ ↔ ∃𝑝[𝐺 / 𝑓]𝜒′) |
18 | 7, 15, 17 | 3bitr4i 304 | . 2 ⊢ ([𝐺 / 𝑓]∃𝑛𝜒 ↔ ∃𝑝𝜒″) |
19 | 6, 18 | bitri 276 | 1 ⊢ (𝐺 ∈ 𝐵 ↔ ∃𝑝𝜒″) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ w3a 1079 = wceq 1528 ∃wex 1771 [wsb 2060 ∈ wcel 2105 {cab 2796 ∃wrex 3136 Vcvv 3492 [wsbc 3769 ∪ cun 3931 {csn 4557 〈cop 4563 Fn wfn 6343 ∧ w-bnj17 31855 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-13 2381 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-nul 4289 df-sn 4558 df-pr 4560 df-uni 4831 df-bnj17 31856 |
This theorem is referenced by: bnj1018 32133 |
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