Theorem bj-projeq2 38033

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 38032	. 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))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   Proj bj-cproj 38030

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 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

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 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-br 5106  df-opab 5170  df-xp 5660  df-cnv 5662  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-bj-proj 38031

This theorem is used by:  bj-pr1eq  38042  bj-pr2eq  38056