Theorem bj-pr1eq 37004

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 36995	. 2 ⊢ (𝐴 = 𝐵 → (∅ Proj 𝐴) = (∅ Proj 𝐵))
2		df-bj-pr1 37003	. 2 ⊢ pr1 𝐴 = (∅ Proj 𝐴)
3		df-bj-pr1 37003	. 2 ⊢ pr1 𝐵 = (∅ Proj 𝐵)
4	1, 2, 3	3eqtr4g 2801	1 ⊢ (𝐴 = 𝐵 → pr1 𝐴 = pr1 𝐵)

Syntax hints:   → wi 4   = wceq 1539  ∅c0 4332   Proj bj-cproj 36992  pr₁ bj-cpr1 37002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-bj-proj 36993  df-bj-pr1 37003

This theorem is referenced by:  bj-pr11val  37007  bj-1uplth  37009  bj-pr21val  37015  bj-2uplth  37023