Theorem bj-pr2eq 37017

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

Step	Hyp	Ref	Expression
1		bj-projeq2 36994	. 2 ⊢ (𝐴 = 𝐵 → (1o Proj 𝐴) = (1o Proj 𝐵))
2		df-bj-pr2 37016	. 2 ⊢ pr2 𝐴 = (1o Proj 𝐴)
3		df-bj-pr2 37016	. 2 ⊢ pr2 𝐵 = (1o Proj 𝐵)
4	1, 2, 3	3eqtr4g 2802	1 ⊢ (𝐴 = 𝐵 → pr2 𝐴 = pr2 𝐵)

Syntax hints:   → wi 4   = wceq 1540  1_oc1o 8499   Proj bj-cproj 36991  pr₂ bj-cpr2 37015

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-bj-proj 36992  df-bj-pr2 37016

This theorem is referenced by:  bj-pr22val  37020  bj-2uplth  37022