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| Mirrors > Home > MPE Home > Th. List > breqan12d | Structured version Visualization version 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 5148 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)) | |
| 4 | 1, 2, 3 | syl2an 596 | 1 ⊢ ((𝜑 ∧ 𝜓) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 class class class wbr 5143 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 |
| This theorem is referenced by: breqan12rd 5160 soisores 7347 isoid 7349 isores3 7355 isoini2 7359 ofrfvalg 7705 fnwelem 8156 fnse 8158 infsupprpr 9544 wemaplem1 9586 r0weon 10052 sornom 10317 enqbreq2 10960 nqereu 10969 ordpinq 10983 lterpq 11010 ltresr2 11181 axpre-ltadd 11207 leltadd 11747 lemul1a 12121 negiso 12248 xltneg 13259 lt2sq 14173 le2sq 14174 expmordi 14207 sqrtle 15299 prdsleval 17522 efgcpbllema 19772 iducn 24292 icopnfhmeo 24974 iccpnfhmeo 24976 xrhmeo 24977 reefiso 26492 sinord 26576 logltb 26642 logccv 26705 atanord 26970 birthdaylem3 26996 lgsquadlem3 27426 sltneg 28077 mddmd 32320 xrge0iifiso 33934 revwlkb 35131 erdszelem4 35199 erdszelem8 35203 satfv0 35363 cgrextend 36009 matunitlindf 37625 idlaut 40098 monotuz 42953 monotoddzzfi 42954 wepwsolem 43054 fnwe2val 43061 aomclem8 43073 isgrlim 47949 rrx2plord 48641 rrx2plordisom 48644 |
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