Theorem bj-tageq 38066

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 38057	. . 3 ⊢ (𝐴 = 𝐵 → sngl 𝐴 = sngl 𝐵)
2	1	uneq1d 4169	. 2 ⊢ (𝐴 = 𝐵 → (sngl 𝐴 ∪ {∅}) = (sngl 𝐵 ∪ {∅}))
3		df-bj-tag 38065	. 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
4		df-bj-tag 38065	. 2 ⊢ tag 𝐵 = (sngl 𝐵 ∪ {∅})
5	2, 3, 4	3eqtr4g 2823	1 ⊢ (𝐴 = 𝐵 → tag 𝐴 = tag 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∪ cun 4152  ∅c0 4281  {csn 4588  sngl bj-csngl 38055  tag bj-ctag 38064

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rex 3091  df-v 3456  df-un 4154  df-bj-sngl 38056  df-bj-tag 38065

This theorem is used by:  bj-xtageq  38078