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Mirrors > Home > MPE Home > Th. List > breqdi | Structured version Visualization version GIF version |
Description: Equality deduction for a binary relation. (Contributed by Thierry Arnoux, 5-Oct-2020.) |
Ref | Expression |
---|---|
breq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
breqdi.1 | ⊢ (𝜑 → 𝐶𝐴𝐷) |
Ref | Expression |
---|---|
breqdi | ⊢ (𝜑 → 𝐶𝐵𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breqdi.1 | . 2 ⊢ (𝜑 → 𝐶𝐴𝐷) | |
2 | breq1d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
3 | 2 | breqd 5159 | . 2 ⊢ (𝜑 → (𝐶𝐴𝐷 ↔ 𝐶𝐵𝐷)) |
4 | 1, 3 | mpbid 231 | 1 ⊢ (𝜑 → 𝐶𝐵𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 class class class wbr 5148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1775 df-cleq 2720 df-clel 2806 df-br 5149 |
This theorem is referenced by: rtrclreclem3 15040 episect 17768 dvef 25925 acopyeu 28651 isleagd 28665 0prjspn 42052 brfvimex 43456 brovmptimex 43457 ntrclsnvobr 43482 clsneibex 43532 neicvgbex 43542 |
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