Theorem bj-pr2eq 38120

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 38097	. 2 ⊢ (𝐴 = 𝐵 → (1o Proj 𝐴) = (1o Proj 𝐵))
2		df-bj-pr2 38119	. 2 ⊢ pr2 𝐴 = (1o Proj 𝐴)
3		df-bj-pr2 38119	. 2 ⊢ pr2 𝐵 = (1o Proj 𝐵)
4	1, 2, 3	3eqtr4g 2823	1 ⊢ (𝐴 = 𝐵 → pr2 𝐴 = pr2 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  1_oc1o 8500   Proj bj-cproj 38094  pr₂ bj-cpr2 38118

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-xp 5662  df-cnv 5664  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-bj-proj 38095  df-bj-pr2 38119

This theorem is used by:  bj-pr22val  38123  bj-2uplth  38125