Theorem ttceqd 36855

Description: Equality deduction for transitive closure. (Contributed by Matthew House, 6-Apr-2026.)

Hypothesis

Ref	Expression
ttceqd.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
ttceqd	⊢ (𝜑 → TC+ 𝐴 = TC+ 𝐵)

Proof of Theorem ttceqd

Step	Hyp	Ref	Expression
1		ttceqd.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		ttceq 36853	. 2 ⊢ (𝐴 = 𝐵 → TC+ 𝐴 = TC+ 𝐵)
3	1, 2	syl 17	1 ⊢ (𝜑 → TC+ 𝐴 = TC+ 𝐵)

Syntax hints:   → wi 4   = wceq 1561  TC+ cttc 36851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1564  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-rex 3088  df-v 3457  df-ss 3922  df-iun 4952  df-ttc 36852

This theorem is referenced by:  csbttc  36874