Theorem bj-pr2eq 37693

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 37670	. 2 ⊢ (𝐴 = 𝐵 → (1o Proj 𝐴) = (1o Proj 𝐵))
2		df-bj-pr2 37692	. 2 ⊢ pr2 𝐴 = (1o Proj 𝐴)
3		df-bj-pr2 37692	. 2 ⊢ pr2 𝐵 = (1o Proj 𝐵)
4	1, 2, 3	3eqtr4g 2826	1 ⊢ (𝐴 = 𝐵 → pr2 𝐴 = pr2 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  1_oc1o 8455   Proj bj-cproj 37667  pr₂ bj-cpr2 37691

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

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 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5672  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-bj-proj 37668  df-bj-pr2 37692

This theorem is used by:  bj-pr22val  37696  bj-2uplth  37698