Theorem bj-projeq2 37649

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 2763	. 2 ⊢ 𝐴 = 𝐴
2		bj-projeq 37648	. 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 1570   Proj bj-cproj 37646

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-xp 5667  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-bj-proj 37647

This theorem is referenced by:  bj-pr1eq  37658  bj-pr2eq  37672