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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj873 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 32284. 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 |
---|---|
bnj873.4 | ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} |
bnj873.7 | ⊢ (𝜑′ ↔ [𝑔 / 𝑓]𝜑) |
bnj873.8 | ⊢ (𝜓′ ↔ [𝑔 / 𝑓]𝜓) |
Ref | Expression |
---|---|
bnj873 | ⊢ 𝐵 = {𝑔 ∣ ∃𝑛 ∈ 𝐷 (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj873.4 | . 2 ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} | |
2 | nfv 1915 | . . 3 ⊢ Ⅎ𝑔∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) | |
3 | nfcv 2979 | . . . 4 ⊢ Ⅎ𝑓𝐷 | |
4 | nfv 1915 | . . . . 5 ⊢ Ⅎ𝑓 𝑔 Fn 𝑛 | |
5 | bnj873.7 | . . . . . 6 ⊢ (𝜑′ ↔ [𝑔 / 𝑓]𝜑) | |
6 | nfsbc1v 3794 | . . . . . 6 ⊢ Ⅎ𝑓[𝑔 / 𝑓]𝜑 | |
7 | 5, 6 | nfxfr 1853 | . . . . 5 ⊢ Ⅎ𝑓𝜑′ |
8 | bnj873.8 | . . . . . 6 ⊢ (𝜓′ ↔ [𝑔 / 𝑓]𝜓) | |
9 | nfsbc1v 3794 | . . . . . 6 ⊢ Ⅎ𝑓[𝑔 / 𝑓]𝜓 | |
10 | 8, 9 | nfxfr 1853 | . . . . 5 ⊢ Ⅎ𝑓𝜓′ |
11 | 4, 7, 10 | nf3an 1902 | . . . 4 ⊢ Ⅎ𝑓(𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′) |
12 | 3, 11 | nfrex 3311 | . . 3 ⊢ Ⅎ𝑓∃𝑛 ∈ 𝐷 (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′) |
13 | fneq1 6446 | . . . . 5 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝑛 ↔ 𝑔 Fn 𝑛)) | |
14 | sbceq1a 3785 | . . . . . 6 ⊢ (𝑓 = 𝑔 → (𝜑 ↔ [𝑔 / 𝑓]𝜑)) | |
15 | 14, 5 | syl6bbr 291 | . . . . 5 ⊢ (𝑓 = 𝑔 → (𝜑 ↔ 𝜑′)) |
16 | sbceq1a 3785 | . . . . . 6 ⊢ (𝑓 = 𝑔 → (𝜓 ↔ [𝑔 / 𝑓]𝜓)) | |
17 | 16, 8 | syl6bbr 291 | . . . . 5 ⊢ (𝑓 = 𝑔 → (𝜓 ↔ 𝜓′)) |
18 | 13, 15, 17 | 3anbi123d 1432 | . . . 4 ⊢ (𝑓 = 𝑔 → ((𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′))) |
19 | 18 | rexbidv 3299 | . . 3 ⊢ (𝑓 = 𝑔 → (∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ∃𝑛 ∈ 𝐷 (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′))) |
20 | 2, 12, 19 | cbvabw 2892 | . 2 ⊢ {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} = {𝑔 ∣ ∃𝑛 ∈ 𝐷 (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)} |
21 | 1, 20 | eqtri 2846 | 1 ⊢ 𝐵 = {𝑔 ∣ ∃𝑛 ∈ 𝐷 (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ w3a 1083 = wceq 1537 {cab 2801 ∃wrex 3141 [wsbc 3774 Fn wfn 6352 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-fun 6359 df-fn 6360 |
This theorem is referenced by: bnj849 32199 bnj893 32202 |
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