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Theorem bj-pr1eq 34589
 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 34580 . 2 (𝐴 = 𝐵 → (∅ Proj 𝐴) = (∅ Proj 𝐵))
2 df-bj-pr1 34588 . 2 pr1 𝐴 = (∅ Proj 𝐴)
3 df-bj-pr1 34588 . 2 pr1 𝐵 = (∅ Proj 𝐵)
41, 2, 33eqtr4g 2858 1 (𝐴 = 𝐵 → pr1 𝐴 = pr1 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538  ∅c0 4246   Proj bj-cproj 34577  pr1 bj-cpr1 34587 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-rab 3115  df-v 3444  df-un 3888  df-in 3890  df-ss 3900  df-sn 4529  df-pr 4531  df-op 4535  df-br 5035  df-opab 5097  df-xp 5529  df-cnv 5531  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-bj-proj 34578  df-bj-pr1 34588 This theorem is referenced by:  bj-pr11val  34592  bj-1uplth  34594  bj-pr21val  34600  bj-2uplth  34608
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