Theorem bj-pr2eq 37384

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 37361	. 2 ⊢ (𝐴 = 𝐵 → (1o Proj 𝐴) = (1o Proj 𝐵))
2		df-bj-pr2 37383	. 2 ⊢ pr2 𝐴 = (1o Proj 𝐴)
3		df-bj-pr2 37383	. 2 ⊢ pr2 𝐵 = (1o Proj 𝐵)
4	1, 2, 3	3eqtr4g 2801	1 ⊢ (𝐴 = 𝐵 → pr2 𝐴 = pr2 𝐵)

Syntax hints:   → wi 4   = wceq 1548  1_oc1o 8392   Proj bj-cproj 37358  pr₂ bj-cpr2 37382

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-xp 5627  df-cnv 5629  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-bj-proj 37359  df-bj-pr2 37383

This theorem is referenced by:  bj-pr22val  37387  bj-2uplth  37389