Theorem bj-tageq 32157

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 32148	. . 3 ⊢ (𝐴 = 𝐵 → sngl 𝐴 = sngl 𝐵)
2	1	uneq1d 3728	. 2 ⊢ (𝐴 = 𝐵 → (sngl 𝐴 ∪ {∅}) = (sngl 𝐵 ∪ {∅}))
3		df-bj-tag 32156	. 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
4		df-bj-tag 32156	. 2 ⊢ tag 𝐵 = (sngl 𝐵 ∪ {∅})
5	2, 3, 4	3eqtr4g 2669	1 ⊢ (𝐴 = 𝐵 → tag 𝐴 = tag 𝐵)

Syntax hints:   → wi 4   = wceq 1475   ∪ cun 3538  ∅c0 3874  {csn 4125  sngl bj-csngl 32146  tag bj-ctag 32155

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-v 3175  df-un 3545  df-bj-sngl 32147  df-bj-tag 32156

This theorem is referenced by:  bj-xtageq  32169