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Mirrors > Home > ILE Home > Th. List > breq123d | GIF version |
Description: Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.) |
Ref | Expression |
---|---|
breq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
breq123d.2 | ⊢ (𝜑 → 𝑅 = 𝑆) |
breq123d.3 | ⊢ (𝜑 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
breq123d | ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑆𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | breq123d.3 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
3 | 1, 2 | breq12d 4000 | . 2 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)) |
4 | breq123d.2 | . . 3 ⊢ (𝜑 → 𝑅 = 𝑆) | |
5 | 4 | breqd 3998 | . 2 ⊢ (𝜑 → (𝐵𝑅𝐷 ↔ 𝐵𝑆𝐷)) |
6 | 3, 5 | bitrd 187 | 1 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑆𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1348 class class class wbr 3987 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-un 3125 df-sn 3587 df-pr 3588 df-op 3590 df-br 3988 |
This theorem is referenced by: sbcbrg 4041 fmptco 5660 |
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