Theorem bj-projeq2 36981

Description: Substitution property for Proj. (Contributed by BJ, 6-Apr-2019.)

Assertion

Ref	Expression
bj-projeq2	⊢ (𝐵 = 𝐶 → (𝐴 Proj 𝐵) = (𝐴 Proj 𝐶))

Proof of Theorem bj-projeq2

Step	Hyp	Ref	Expression
1		eqid 2729	. 2 ⊢ 𝐴 = 𝐴
2		bj-projeq 36980	. 2 ⊢ (𝐴 = 𝐴 → (𝐵 = 𝐶 → (𝐴 Proj 𝐵) = (𝐴 Proj 𝐶)))
3	1, 2	ax-mp 5	1 ⊢ (𝐵 = 𝐶 → (𝐴 Proj 𝐵) = (𝐴 Proj 𝐶))

Syntax hints:   → wi 4   = wceq 1540   Proj bj-cproj 36978

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 2701

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 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-bj-proj 36979

This theorem is referenced by:  bj-pr1eq  36990  bj-pr2eq  37004