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Mirrors > Home > ILE Home > Th. List > sbcbr1g | GIF version |
Description: Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) |
Ref | Expression |
---|---|
sbcbr1g | ⊢ (𝐴 ∈ 𝐷 → ([𝐴 / 𝑥]𝐵𝑅𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵𝑅𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcbr12g 3835 | . 2 ⊢ (𝐴 ∈ 𝐷 → ([𝐴 / 𝑥]𝐵𝑅𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵𝑅⦋𝐴 / 𝑥⦌𝐶)) | |
2 | csbconstg 2920 | . . 3 ⊢ (𝐴 ∈ 𝐷 → ⦋𝐴 / 𝑥⦌𝐶 = 𝐶) | |
3 | 2 | breq2d 3797 | . 2 ⊢ (𝐴 ∈ 𝐷 → (⦋𝐴 / 𝑥⦌𝐵𝑅⦋𝐴 / 𝑥⦌𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵𝑅𝐶)) |
4 | 1, 3 | bitrd 186 | 1 ⊢ (𝐴 ∈ 𝐷 → ([𝐴 / 𝑥]𝐵𝑅𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵𝑅𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∈ wcel 1433 [wsbc 2815 ⦋csb 2908 class class class wbr 3785 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-sbc 2816 df-csb 2909 df-un 2977 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 |
This theorem is referenced by: (None) |
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