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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrneq2i | 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 |
---|---|
xrneq2i.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
xrneq2i | ⊢ (𝐶 ⋉ 𝐴) = (𝐶 ⋉ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrneq2i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | xrneq2 38338 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 ⋉ 𝐴) = (𝐶 ⋉ 𝐵)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐶 ⋉ 𝐴) = (𝐶 ⋉ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ⋉ cxrn 38136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2711 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1778 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-rab 3444 df-in 3983 df-ss 3993 df-br 5167 df-opab 5229 df-co 5709 df-xrn 38329 |
This theorem is referenced by: disjsuc 38717 |
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