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Theorem bj-pr1eq 36186
Description: Substitution property for pr1. (Contributed by BJ, 6-Apr-2019.)
Assertion
Ref Expression
bj-pr1eq (𝐴 = 𝐵 → pr1 𝐴 = pr1 𝐵)

Proof of Theorem bj-pr1eq
StepHypRef Expression
1 bj-projeq2 36177 . 2 (𝐴 = 𝐵 → (∅ Proj 𝐴) = (∅ Proj 𝐵))
2 df-bj-pr1 36185 . 2 pr1 𝐴 = (∅ Proj 𝐴)
3 df-bj-pr1 36185 . 2 pr1 𝐵 = (∅ Proj 𝐵)
41, 2, 33eqtr4g 2795 1 (𝐴 = 𝐵 → pr1 𝐴 = pr1 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  c0 4321   Proj bj-cproj 36174  pr1 bj-cpr1 36184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-bj-proj 36175  df-bj-pr1 36185
This theorem is referenced by:  bj-pr11val  36189  bj-1uplth  36191  bj-pr21val  36197  bj-2uplth  36205
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