Theorem ttceqd 37456

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 37454	. 2 ⊢ (𝐴 = 𝐵 → TC+ 𝐴 = TC+ 𝐵)
3	1, 2	syl 18	1 ⊢ (𝜑 → TC+ 𝐴 = TC+ 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  TC+ cttc 37452

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-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rex 3091  df-v 3456  df-ss 3901  df-iun 4956  df-ttc 37453

This theorem is used by:  csbttc  37475