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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrneq12d | Structured version Visualization version GIF version |
Description: Equality theorem for the range Cartesian product, deduction form. (Contributed by Peter Mazsa, 18-Dec-2021.) |
Ref | Expression |
---|---|
xrneq12d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
xrneq12d.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
xrneq12d | ⊢ (𝜑 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrneq12d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | xrneq12d.2 | . 2 ⊢ (𝜑 → 𝐶 = 𝐷) | |
3 | xrneq12 37765 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷)) | |
4 | 1, 2, 3 | syl2anc 583 | 1 ⊢ (𝜑 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ⋉ cxrn 37554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-rab 3427 df-v 3470 df-in 3950 df-ss 3960 df-br 5142 df-opab 5204 df-co 5678 df-xrn 37753 |
This theorem is referenced by: (None) |
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