Theorem bj-pr1eq 35685

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 35676	. 2 ⊢ (𝐴 = 𝐵 → (∅ Proj 𝐴) = (∅ Proj 𝐵))
2		df-bj-pr1 35684	. 2 ⊢ pr1 𝐴 = (∅ Proj 𝐴)
3		df-bj-pr1 35684	. 2 ⊢ pr1 𝐵 = (∅ Proj 𝐵)
4	1, 2, 3	3eqtr4g 2796	1 ⊢ (𝐴 = 𝐵 → pr1 𝐴 = pr1 𝐵)

Syntax hints:   → wi 4   = wceq 1541  ∅c0 4318   Proj bj-cproj 35673  pr₁ bj-cpr1 35683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-xp 5675  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-bj-proj 35674  df-bj-pr1 35684

This theorem is referenced by:  bj-pr11val  35688  bj-1uplth  35690  bj-pr21val  35696  bj-2uplth  35704