Theorem shiftstableeq2 39057

Description: Equality theorem for shift-stability of two classes. (Contributed by Peter Mazsa, 19-Feb-2026.)

Assertion

Ref	Expression
shiftstableeq2	⊢ (𝐹 = 𝐺 → (𝑆 ShiftStable 𝐹) = (𝑆 ShiftStable 𝐺))

Proof of Theorem shiftstableeq2

Step	Hyp	Ref	Expression
1		coeq2 5845	. . 3 ⊢ (𝐹 = 𝐺 → (𝑆 ∘ 𝐹) = (𝑆 ∘ 𝐺))
2		id 23	. . 3 ⊢ (𝐹 = 𝐺 → 𝐹 = 𝐺)
3	1, 2	ineq12d 4180	. 2 ⊢ (𝐹 = 𝐺 → ((𝑆 ∘ 𝐹) ∩ 𝐹) = ((𝑆 ∘ 𝐺) ∩ 𝐺))
4		df-shiftstable 39056	. 2 ⊢ (𝑆 ShiftStable 𝐹) = ((𝑆 ∘ 𝐹) ∩ 𝐹)
5		df-shiftstable 39056	. 2 ⊢ (𝑆 ShiftStable 𝐺) = ((𝑆 ∘ 𝐺) ∩ 𝐺)
6	3, 4, 5	3eqtr4g 2829	1 ⊢ (𝐹 = 𝐺 → (𝑆 ShiftStable 𝐹) = (𝑆 ShiftStable 𝐺))

Syntax hints:   → wi 4   = wceq 1567   ∩ cin 3910   ∘ ccom 5666   ShiftStable cshiftstable 38756

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3423  df-in 3918  df-ss 3928  df-br 5112  df-opab 5176  df-co 5671  df-shiftstable 39056

This theorem is referenced by:  dfpet2parts2  39547  pets2eq  39551