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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrneq1i | Structured version Visualization version GIF version |
Description: Equality theorem for the range Cartesian product, inference form. (Contributed by Peter Mazsa, 16-Dec-2020.) |
Ref | Expression |
---|---|
xrneq1i.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
xrneq1i | ⊢ (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrneq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | xrneq1 35663 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ⋉ cxrn 35486 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rab 3146 df-in 3936 df-ss 3945 df-br 5060 df-opab 5122 df-co 5557 df-xrn 35657 |
This theorem is referenced by: (None) |
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