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Mirrors > Home > MPE Home > Th. List > sbcbr123 | Structured version Visualization version GIF version |
Description: Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Revised by NM, 22-Aug-2018.) |
Ref | Expression |
---|---|
sbcbr123 | ⊢ ([𝐴 / 𝑥]𝐵𝑅𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝑅⦋𝐴 / 𝑥⦌𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcex 3716 | . 2 ⊢ ([𝐴 / 𝑥]𝐵𝑅𝐶 → 𝐴 ∈ V) | |
2 | br0 5011 | . . . 4 ⊢ ¬ ⦋𝐴 / 𝑥⦌𝐵∅⦋𝐴 / 𝑥⦌𝐶 | |
3 | csbprc 4278 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝑅 = ∅) | |
4 | 3 | breqd 4973 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝑅⦋𝐴 / 𝑥⦌𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵∅⦋𝐴 / 𝑥⦌𝐶)) |
5 | 2, 4 | mtbiri 328 | . . 3 ⊢ (¬ 𝐴 ∈ V → ¬ ⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝑅⦋𝐴 / 𝑥⦌𝐶) |
6 | 5 | con4i 114 | . 2 ⊢ (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝑅⦋𝐴 / 𝑥⦌𝐶 → 𝐴 ∈ V) |
7 | dfsbcq2 3709 | . . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝐵𝑅𝐶 ↔ [𝐴 / 𝑥]𝐵𝑅𝐶)) | |
8 | csbeq1 3814 | . . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵) | |
9 | csbeq1 3814 | . . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝑅 = ⦋𝐴 / 𝑥⦌𝑅) | |
10 | csbeq1 3814 | . . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶) | |
11 | 8, 9, 10 | breq123d 4976 | . . 3 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝑅⦋𝑦 / 𝑥⦌𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝑅⦋𝐴 / 𝑥⦌𝐶)) |
12 | nfcsb1v 3833 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
13 | nfcsb1v 3833 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝑅 | |
14 | nfcsb1v 3833 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶 | |
15 | 12, 13, 14 | nfbr 5009 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝑅⦋𝑦 / 𝑥⦌𝐶 |
16 | csbeq1a 3824 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
17 | csbeq1a 3824 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝑅 = ⦋𝑦 / 𝑥⦌𝑅) | |
18 | csbeq1a 3824 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶) | |
19 | 16, 17, 18 | breq123d 4976 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐵𝑅𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝑅⦋𝑦 / 𝑥⦌𝐶)) |
20 | 15, 19 | sbie 2498 | . . 3 ⊢ ([𝑦 / 𝑥]𝐵𝑅𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝑅⦋𝑦 / 𝑥⦌𝐶) |
21 | 7, 11, 20 | vtoclbg 3511 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝑅𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝑅⦋𝐴 / 𝑥⦌𝐶)) |
22 | 1, 6, 21 | pm5.21nii 380 | 1 ⊢ ([𝐴 / 𝑥]𝐵𝑅𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝑅⦋𝐴 / 𝑥⦌𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 207 = wceq 1522 [wsb 2042 ∈ wcel 2081 Vcvv 3437 [wsbc 3706 ⦋csb 3811 ∅c0 4211 class class class wbr 4962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-br 4963 |
This theorem is referenced by: sbcbr 5017 sbcbr12g 5018 csbcnvgALT 5641 sbcfung 6249 csbfv12 6581 relowlpssretop 34195 |
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