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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 33852 | . 2 ⊢ (𝐴 = 𝐵 → (1o Proj 𝐴) = (1o Proj 𝐵)) | |
2 | df-bj-pr2 33874 | . 2 ⊢ pr2 𝐴 = (1o Proj 𝐴) | |
3 | df-bj-pr2 33874 | . 2 ⊢ pr2 𝐵 = (1o Proj 𝐵) | |
4 | 1, 2, 3 | 3eqtr4g 2833 | 1 ⊢ (𝐴 = 𝐵 → pr2 𝐴 = pr2 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 1oc1o 7896 Proj bj-cproj 33849 pr2 bj-cpr2 33873 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2744 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-rab 3091 df-v 3411 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-br 4926 df-opab 4988 df-xp 5409 df-cnv 5411 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-bj-proj 33850 df-bj-pr2 33874 |
This theorem is referenced by: bj-pr22val 33878 bj-2uplth 33880 |
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