Theorem shiftstableeq2 38865

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 5803	. . 3 ⊢ (𝐹 = 𝐺 → (𝑆 ∘ 𝐹) = (𝑆 ∘ 𝐺))
2		id 22	. . 3 ⊢ (𝐹 = 𝐺 → 𝐹 = 𝐺)
3	1, 2	ineq12d 4153	. 2 ⊢ (𝐹 = 𝐺 → ((𝑆 ∘ 𝐹) ∩ 𝐹) = ((𝑆 ∘ 𝐺) ∩ 𝐺))
4		df-shiftstable 38864	. 2 ⊢ (𝑆 ShiftStable 𝐹) = ((𝑆 ∘ 𝐹) ∩ 𝐹)
5		df-shiftstable 38864	. 2 ⊢ (𝑆 ShiftStable 𝐺) = ((𝑆 ∘ 𝐺) ∩ 𝐺)
6	3, 4, 5	3eqtr4g 2801	1 ⊢ (𝐹 = 𝐺 → (𝑆 ShiftStable 𝐹) = (𝑆 ShiftStable 𝐺))

Syntax hints:   → wi 4   = wceq 1548   ∩ cin 3884   ∘ ccom 5625   ShiftStable cshiftstable 38564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-rab 3394  df-in 3892  df-ss 3902  df-br 5076  df-opab 5138  df-co 5630  df-shiftstable 38864

This theorem is referenced by:  dfpet2parts2  39355  pets2eq  39359