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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 4815 | . 2 ⊢ (𝜑 → (𝐶𝐴𝐷 ↔ 𝐶𝐵𝐷)) |
4 | 1, 3 | mpbid 222 | 1 ⊢ (𝜑 → 𝐶𝐵𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1632 class class class wbr 4804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-an 385 df-ex 1854 df-cleq 2753 df-clel 2756 df-br 4805 |
This theorem is referenced by: brfvimex 38826 brovmptimex 38827 ntrclsnvobr 38852 clsneibex 38902 neicvgbex 38912 |
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