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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj985 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 33679. 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). See bnj985v 33622 for a version with more disjoint variable conditions, not requiring ax-13 2371. (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 33435 | . . 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 33621 | . . 3 ⊢ (𝐺 ∈ V → (𝐺 ∈ 𝐵 ↔ [𝐺 / 𝑓]∃𝑛𝜒)) |
6 | 2, 5 | ax-mp 5 | . 2 ⊢ (𝐺 ∈ 𝐵 ↔ [𝐺 / 𝑓]∃𝑛𝜒) |
7 | sbcex2 3805 | . . 3 ⊢ ([𝐺 / 𝑓]∃𝑝𝜒′ ↔ ∃𝑝[𝐺 / 𝑓]𝜒′) | |
8 | nfv 1918 | . . . . . . 7 ⊢ Ⅎ𝑝𝜒 | |
9 | 8 | sb8e 2517 | . . . . . 6 ⊢ (∃𝑛𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒) |
10 | sbsbc 3744 | . . . . . . 7 ⊢ ([𝑝 / 𝑛]𝜒 ↔ [𝑝 / 𝑛]𝜒) | |
11 | 10 | exbii 1851 | . . . . . 6 ⊢ (∃𝑝[𝑝 / 𝑛]𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒) |
12 | 9, 11 | bitri 275 | . . . . 5 ⊢ (∃𝑛𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒) |
13 | bnj985.6 | . . . . 5 ⊢ (𝜒′ ↔ [𝑝 / 𝑛]𝜒) | |
14 | 12, 13 | bnj133 33396 | . . . 4 ⊢ (∃𝑛𝜒 ↔ ∃𝑝𝜒′) |
15 | 14 | sbcbii 3800 | . . 3 ⊢ ([𝐺 / 𝑓]∃𝑛𝜒 ↔ [𝐺 / 𝑓]∃𝑝𝜒′) |
16 | bnj985.9 | . . . 4 ⊢ (𝜒″ ↔ [𝐺 / 𝑓]𝜒′) | |
17 | 16 | exbii 1851 | . . 3 ⊢ (∃𝑝𝜒″ ↔ ∃𝑝[𝐺 / 𝑓]𝜒′) |
18 | 7, 15, 17 | 3bitr4i 303 | . 2 ⊢ ([𝐺 / 𝑓]∃𝑛𝜒 ↔ ∃𝑝𝜒″) |
19 | 6, 18 | bitri 275 | 1 ⊢ (𝐺 ∈ 𝐵 ↔ ∃𝑝𝜒″) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ w3a 1088 = wceq 1542 ∃wex 1782 [wsb 2068 ∈ wcel 2107 {cab 2710 ∃wrex 3070 Vcvv 3444 [wsbc 3740 ∪ cun 3909 {csn 4587 ⟨cop 4593 Fn wfn 6492 ∧ w-bnj17 33355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-13 2371 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rex 3071 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-sn 4588 df-pr 4590 df-uni 4867 df-bnj17 33356 |
This theorem is referenced by: bnj1018g 33632 |
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