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Mirrors > Home > MPE Home > Th. List > breqan12rd | 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 |
---|---|
breqan12rd | ⊢ ((𝜓 ∧ 𝜑) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | breqan12i.2 | . . 3 ⊢ (𝜓 → 𝐶 = 𝐷) | |
3 | 1, 2 | breqan12d 5056 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)) |
4 | 3 | ancoms 462 | 1 ⊢ ((𝜓 ∧ 𝜑) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 class class class wbr 5040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2711 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-v 3402 df-un 3858 df-sn 4527 df-pr 4529 df-op 4533 df-br 5041 |
This theorem is referenced by: f1oweALT 7711 ledivdiv 11620 xltnegi 12705 ramub1lem1 16475 dvferm1 24750 dvferm2 24752 dvivthlem1 24773 ulmdvlem3 25162 gausslemma2dlem3 26117 lgsquad 26132 areacirclem4 35524 areacirclem5 35525 iccpartgt 44461 |
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