Theorem bj-pr1eq 37990

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 37981	. 2 ⊢ (𝐴 = 𝐵 → (∅ Proj 𝐴) = (∅ Proj 𝐵))
2		df-bj-pr1 37989	. 2 ⊢ pr1 𝐴 = (∅ Proj 𝐴)
3		df-bj-pr1 37989	. 2 ⊢ pr1 𝐵 = (∅ Proj 𝐵)
4	1, 2, 3	3eqtr4g 2820	1 ⊢ (𝐴 = 𝐵 → pr1 𝐴 = pr1 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1569  ∅c0 4278   Proj bj-cproj 37978  pr₁ bj-cpr1 37988

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5656  df-cnv 5658  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-bj-proj 37979  df-bj-pr1 37989

This theorem is used by:  bj-pr11val  37993  bj-1uplth  37995  bj-pr21val  38001  bj-2uplth  38009