Theorem bj-tageq 37535

Description: Substitution property for tag. (Contributed by BJ, 6-Oct-2018.)

Assertion

Ref	Expression
bj-tageq	⊢ (𝐴 = 𝐵 → tag 𝐴 = tag 𝐵)

Proof of Theorem bj-tageq

Step	Hyp	Ref	Expression
1		bj-sngleq 37526	. . 3 ⊢ (𝐴 = 𝐵 → sngl 𝐴 = sngl 𝐵)
2	1	uneq1d 4127	. 2 ⊢ (𝐴 = 𝐵 → (sngl 𝐴 ∪ {∅}) = (sngl 𝐵 ∪ {∅}))
3		df-bj-tag 37534	. 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
4		df-bj-tag 37534	. 2 ⊢ tag 𝐵 = (sngl 𝐵 ∪ {∅})
5	2, 3, 4	3eqtr4g 2829	1 ⊢ (𝐴 = 𝐵 → tag 𝐴 = tag 𝐵)

Syntax hints:   → wi 4   = wceq 1567   ∪ cun 3909  ∅c0 4292  {csn 4592  sngl bj-csngl 37524  tag bj-ctag 37533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rex 3096  df-v 3463  df-un 3916  df-bj-sngl 37525  df-bj-tag 37534

This theorem is referenced by:  bj-xtageq  37547