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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-xtageq | Structured version Visualization version GIF version | ||
| Description: The products of a given class and the tagging of either of two equal classes are equal. (Contributed by BJ, 6-Apr-2019.) | 
| Ref | Expression | 
|---|---|
| bj-xtageq | ⊢ (𝐴 = 𝐵 → (𝐶 × tag 𝐴) = (𝐶 × tag 𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | bj-tageq 36978 | . 2 ⊢ (𝐴 = 𝐵 → tag 𝐴 = tag 𝐵) | |
| 2 | 1 | xpeq2d 5714 | 1 ⊢ (𝐴 = 𝐵 → (𝐶 × tag 𝐴) = (𝐶 × tag 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 × cxp 5682 tag bj-ctag 36976 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rex 3070 df-v 3481 df-un 3955 df-opab 5205 df-xp 5690 df-bj-sngl 36968 df-bj-tag 36977 | 
| This theorem is referenced by: bj-1upleq 37001 bj-2upleq 37014 | 
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