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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-pr2eq | Structured version Visualization version GIF version |
Description: Substitution property for pr2. (Contributed by BJ, 6-Oct-2018.) |
Ref | Expression |
---|---|
bj-pr2eq | ⊢ (𝐴 = 𝐵 → pr2 𝐴 = pr2 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-projeq2 36338 | . 2 ⊢ (𝐴 = 𝐵 → (1o Proj 𝐴) = (1o Proj 𝐵)) | |
2 | df-bj-pr2 36360 | . 2 ⊢ pr2 𝐴 = (1o Proj 𝐴) | |
3 | df-bj-pr2 36360 | . 2 ⊢ pr2 𝐵 = (1o Proj 𝐵) | |
4 | 1, 2, 3 | 3eqtr4g 2796 | 1 ⊢ (𝐴 = 𝐵 → pr2 𝐴 = pr2 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 1oc1o 8465 Proj bj-cproj 36335 pr2 bj-cpr2 36359 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-cnv 5684 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-bj-proj 36336 df-bj-pr2 36360 |
This theorem is referenced by: bj-pr22val 36364 bj-2uplth 36366 |
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