Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-pr2eq Structured version   Visualization version   GIF version

Theorem bj-pr2eq 33875
Description: Substitution property for pr2. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-pr2eq (𝐴 = 𝐵 → pr2 𝐴 = pr2 𝐵)

Proof of Theorem bj-pr2eq
StepHypRef Expression
1 bj-projeq2 33852 . 2 (𝐴 = 𝐵 → (1o Proj 𝐴) = (1o Proj 𝐵))
2 df-bj-pr2 33874 . 2 pr2 𝐴 = (1o Proj 𝐴)
3 df-bj-pr2 33874 . 2 pr2 𝐵 = (1o Proj 𝐵)
41, 2, 33eqtr4g 2833 1 (𝐴 = 𝐵 → pr2 𝐴 = pr2 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1507  1oc1o 7896   Proj bj-cproj 33849  pr2 bj-cpr2 33873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2744
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-rab 3091  df-v 3411  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4926  df-opab 4988  df-xp 5409  df-cnv 5411  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-bj-proj 33850  df-bj-pr2 33874
This theorem is referenced by:  bj-pr22val  33878  bj-2uplth  33880
  Copyright terms: Public domain W3C validator