Theorem bj-pr1eq 37025

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

Step	Hyp	Ref	Expression
1		bj-projeq2 37016	. 2 ⊢ (𝐴 = 𝐵 → (∅ Proj 𝐴) = (∅ Proj 𝐵))
2		df-bj-pr1 37024	. 2 ⊢ pr1 𝐴 = (∅ Proj 𝐴)
3		df-bj-pr1 37024	. 2 ⊢ pr1 𝐵 = (∅ Proj 𝐵)
4	1, 2, 3	3eqtr4g 2796	1 ⊢ (𝐴 = 𝐵 → pr1 𝐴 = pr1 𝐵)

Syntax hints:   → wi 4   = wceq 1540  ∅c0 4313   Proj bj-cproj 37013  pr₁ bj-cpr1 37023

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 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-xp 5665  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-bj-proj 37014  df-bj-pr1 37024

This theorem is referenced by:  bj-pr11val  37028  bj-1uplth  37030  bj-pr21val  37036  bj-2uplth  37044