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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-projeq2 | Structured version Visualization version GIF version |
Description: Substitution property for Proj. (Contributed by BJ, 6-Apr-2019.) |
Ref | Expression |
---|---|
bj-projeq2 | ⊢ (𝐵 = 𝐶 → (𝐴 Proj 𝐵) = (𝐴 Proj 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . 2 ⊢ 𝐴 = 𝐴 | |
2 | bj-projeq 34827 | . 2 ⊢ (𝐴 = 𝐴 → (𝐵 = 𝐶 → (𝐴 Proj 𝐵) = (𝐴 Proj 𝐶))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐵 = 𝐶 → (𝐴 Proj 𝐵) = (𝐴 Proj 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 Proj bj-cproj 34825 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-rab 3062 df-v 3400 df-un 3848 df-in 3850 df-ss 3860 df-sn 4517 df-pr 4519 df-op 4523 df-br 5031 df-opab 5093 df-xp 5531 df-cnv 5533 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-bj-proj 34826 |
This theorem is referenced by: bj-pr1eq 34837 bj-pr2eq 34851 |
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