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Mirrors > Home > ILE Home > Th. List > breqan12d | GIF version |
Description: Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.) |
Ref | Expression |
---|---|
breq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
breqan12i.2 | ⊢ (𝜓 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
breqan12d | ⊢ ((𝜑 ∧ 𝜓) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | breqan12i.2 | . 2 ⊢ (𝜓 → 𝐶 = 𝐷) | |
3 | breq12 3872 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)) | |
4 | 1, 2, 3 | syl2an 284 | 1 ⊢ ((𝜑 ∧ 𝜓) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1296 class class class wbr 3867 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-v 2635 df-un 3017 df-sn 3472 df-pr 3473 df-op 3475 df-br 3868 |
This theorem is referenced by: breqan12rd 3883 sosng 4540 isoresbr 5626 isoid 5627 isores3 5632 isoini2 5636 ofrfval 5902 oviec 6438 enqbreq2 7013 ltresr2 7474 axpre-ltadd 7518 leltadd 8022 xltneg 9402 lt2sq 10159 le2sq 10160 sqrtle 10600 sqrtlt 10601 absext 10627 |
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