Theorem bj-pr2eq 37039

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 37016	. 2 ⊢ (𝐴 = 𝐵 → (1o Proj 𝐴) = (1o Proj 𝐵))
2		df-bj-pr2 37038	. 2 ⊢ pr2 𝐴 = (1o Proj 𝐴)
3		df-bj-pr2 37038	. 2 ⊢ pr2 𝐵 = (1o Proj 𝐵)
4	1, 2, 3	3eqtr4g 2796	1 ⊢ (𝐴 = 𝐵 → pr2 𝐴 = pr2 𝐵)

Syntax hints:   → wi 4   = wceq 1540  1_oc1o 8478   Proj bj-cproj 37013  pr₂ bj-cpr2 37037

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-pr2 37038

This theorem is referenced by:  bj-pr22val  37042  bj-2uplth  37044