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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrneq1d | Structured version Visualization version GIF version |
Description: Equality theorem for the range Cartesian product, deduction form. (Contributed by Peter Mazsa, 7-Sep-2021.) |
Ref | Expression |
---|---|
xrneq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
xrneq1d | ⊢ (𝜑 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrneq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | xrneq1 36413 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ⋉ cxrn 36238 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2717 df-cleq 2731 df-clel 2818 df-rab 3073 df-v 3425 df-in 3891 df-ss 3901 df-br 5071 df-opab 5133 df-co 5588 df-xrn 36407 |
This theorem is referenced by: (None) |
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