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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-pr1eq | Structured version Visualization version GIF version |
Description: Substitution property for pr1. (Contributed by BJ, 6-Apr-2019.) |
Ref | Expression |
---|---|
bj-pr1eq | ⊢ (𝐴 = 𝐵 → pr1 𝐴 = pr1 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-projeq2 36976 | . 2 ⊢ (𝐴 = 𝐵 → (∅ Proj 𝐴) = (∅ Proj 𝐵)) | |
2 | df-bj-pr1 36984 | . 2 ⊢ pr1 𝐴 = (∅ Proj 𝐴) | |
3 | df-bj-pr1 36984 | . 2 ⊢ pr1 𝐵 = (∅ Proj 𝐵) | |
4 | 1, 2, 3 | 3eqtr4g 2800 | 1 ⊢ (𝐴 = 𝐵 → pr1 𝐴 = pr1 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∅c0 4339 Proj bj-cproj 36973 pr1 bj-cpr1 36983 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-bj-proj 36974 df-bj-pr1 36984 |
This theorem is referenced by: bj-pr11val 36988 bj-1uplth 36990 bj-pr21val 36996 bj-2uplth 37004 |
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