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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 35631 | . 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 1537 ⋉ cxrn 35454 |
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 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-in 3945 df-ss 3954 df-br 5069 df-opab 5131 df-co 5566 df-xrn 35625 |
This theorem is referenced by: (None) |
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