Theorem bj-projeq2 34828

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 2738	. 2 ⊢ 𝐴 = 𝐴
2		bj-projeq 34827	. 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 1542   Proj bj-cproj 34825

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-rab 3062  df-v 3400  df-un 3848  df-in 3850  df-ss 3860  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-xp 5531  df-cnv 5533  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-bj-proj 34826

This theorem is referenced by:  bj-pr1eq  34837  bj-pr2eq  34851