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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 34300 | . 2 ⊢ (𝐴 = 𝐵 → (1o Proj 𝐴) = (1o Proj 𝐵)) | |
2 | df-bj-pr2 34322 | . 2 ⊢ pr2 𝐴 = (1o Proj 𝐴) | |
3 | df-bj-pr2 34322 | . 2 ⊢ pr2 𝐵 = (1o Proj 𝐵) | |
4 | 1, 2, 3 | 3eqtr4g 2881 | 1 ⊢ (𝐴 = 𝐵 → pr2 𝐴 = pr2 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 1oc1o 8089 Proj bj-cproj 34297 pr2 bj-cpr2 34321 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-xp 5555 df-cnv 5557 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-bj-proj 34298 df-bj-pr2 34322 |
This theorem is referenced by: bj-pr22val 34326 bj-2uplth 34328 |
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