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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 5071 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)) | |
4 | 1, 2, 3 | syl2an 597 | 1 ⊢ ((𝜑 ∧ 𝜓) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 class class class wbr 5066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 |
This theorem is referenced by: breqan12rd 5083 soisores 7080 isoid 7082 isores3 7088 isoini2 7092 ofrfval 7417 fnwelem 7825 fnse 7827 infsupprpr 8968 wemaplem1 9010 r0weon 9438 sornom 9699 enqbreq2 10342 nqereu 10351 ordpinq 10365 lterpq 10392 ltresr2 10563 axpre-ltadd 10589 leltadd 11124 lemul1a 11494 negiso 11621 xltneg 12611 lt2sq 13499 le2sq 13500 expmordi 13532 sqrtle 14620 prdsleval 16750 efgcpbllema 18880 iducn 22892 icopnfhmeo 23547 iccpnfhmeo 23549 xrhmeo 23550 reefiso 25036 sinord 25118 logltb 25183 logccv 25246 atanord 25505 birthdaylem3 25531 lgsquadlem3 25958 mddmd 30078 xrge0iifiso 31178 revwlkb 32372 erdszelem4 32441 erdszelem8 32445 satfv0 32605 cgrextend 33469 matunitlindf 34905 idlaut 37247 monotuz 39558 monotoddzzfi 39559 wepwsolem 39662 fnwe2val 39669 aomclem8 39681 rrx2plord 44727 rrx2plordisom 44730 |
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