Theorem bj-pr2eq 33004

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 32981	. 2 ⊢ (𝐴 = 𝐵 → (1𝑜 Proj 𝐴) = (1𝑜 Proj 𝐵))
2		df-bj-pr2 33003	. 2 ⊢ pr2 𝐴 = (1𝑜 Proj 𝐴)
3		df-bj-pr2 33003	. 2 ⊢ pr2 𝐵 = (1𝑜 Proj 𝐵)
4	1, 2, 3	3eqtr4g 2681	1 ⊢ (𝐴 = 𝐵 → pr2 𝐴 = pr2 𝐵)

Syntax hints:   → wi 4   = wceq 1483  1_𝑜c1o 7553   Proj bj-cproj 32978  pr₂ bj-cpr2 33002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-bj-proj 32979  df-bj-pr2 33003

This theorem is referenced by:  bj-pr22val  33007  bj-2uplth  33009