Theorem bj-tageq 37150

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 37141	. . 3 ⊢ (𝐴 = 𝐵 → sngl 𝐴 = sngl 𝐵)
2	1	uneq1d 4118	. 2 ⊢ (𝐴 = 𝐵 → (sngl 𝐴 ∪ {∅}) = (sngl 𝐵 ∪ {∅}))
3		df-bj-tag 37149	. 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
4		df-bj-tag 37149	. 2 ⊢ tag 𝐵 = (sngl 𝐵 ∪ {∅})
5	2, 3, 4	3eqtr4g 2795	1 ⊢ (𝐴 = 𝐵 → tag 𝐴 = tag 𝐵)

Syntax hints:   → wi 4   = wceq 1542   ∪ cun 3898  ∅c0 4284  {csn 4579  sngl bj-csngl 37139  tag bj-ctag 37148

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-rex 3060  df-v 3441  df-un 3905  df-bj-sngl 37140  df-bj-tag 37149

This theorem is referenced by:  bj-xtageq  37162