Theorem bj-pr1eq 36968

Description: Substitution property for pr₁. (Contributed by BJ, 6-Apr-2019.)

Assertion

Ref	Expression
bj-pr1eq	⊢ (𝐴 = 𝐵 → pr1 𝐴 = pr1 𝐵)

Proof of Theorem bj-pr1eq

Step	Hyp	Ref	Expression
1		bj-projeq2 36959	. 2 ⊢ (𝐴 = 𝐵 → (∅ Proj 𝐴) = (∅ Proj 𝐵))
2		df-bj-pr1 36967	. 2 ⊢ pr1 𝐴 = (∅ Proj 𝐴)
3		df-bj-pr1 36967	. 2 ⊢ pr1 𝐵 = (∅ Proj 𝐵)
4	1, 2, 3	3eqtr4g 2805	1 ⊢ (𝐴 = 𝐵 → pr1 𝐴 = pr1 𝐵)

Syntax hints:   → wi 4   = wceq 1537  ∅c0 4352   Proj bj-cproj 36956  pr₁ bj-cpr1 36966

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-bj-proj 36957  df-bj-pr1 36967

This theorem is referenced by:  bj-pr11val  36971  bj-1uplth  36973  bj-pr21val  36979  bj-2uplth  36987