Theorem bj-tageq 36971

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 36962	. . 3 ⊢ (𝐴 = 𝐵 → sngl 𝐴 = sngl 𝐵)
2	1	uneq1d 4133	. 2 ⊢ (𝐴 = 𝐵 → (sngl 𝐴 ∪ {∅}) = (sngl 𝐵 ∪ {∅}))
3		df-bj-tag 36970	. 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
4		df-bj-tag 36970	. 2 ⊢ tag 𝐵 = (sngl 𝐵 ∪ {∅})
5	2, 3, 4	3eqtr4g 2790	1 ⊢ (𝐴 = 𝐵 → tag 𝐴 = tag 𝐵)

Syntax hints:   → wi 4   = wceq 1540   ∪ cun 3915  ∅c0 4299  {csn 4592  sngl bj-csngl 36960  tag bj-ctag 36969

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rex 3055  df-v 3452  df-un 3922  df-bj-sngl 36961  df-bj-tag 36970

This theorem is referenced by:  bj-xtageq  36983