Theorem shiftstableeq2 38987

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 5832	. . 3 ⊢ (𝐹 = 𝐺 → (𝑆 ∘ 𝐹) = (𝑆 ∘ 𝐺))
2		id 22	. . 3 ⊢ (𝐹 = 𝐺 → 𝐹 = 𝐺)
3	1, 2	ineq12d 4175	. 2 ⊢ (𝐹 = 𝐺 → ((𝑆 ∘ 𝐹) ∩ 𝐹) = ((𝑆 ∘ 𝐺) ∩ 𝐺))
4		df-shiftstable 38986	. 2 ⊢ (𝑆 ShiftStable 𝐹) = ((𝑆 ∘ 𝐹) ∩ 𝐹)
5		df-shiftstable 38986	. 2 ⊢ (𝑆 ShiftStable 𝐺) = ((𝑆 ∘ 𝐺) ∩ 𝐺)
6	3, 4, 5	3eqtr4g 2824	1 ⊢ (𝐹 = 𝐺 → (𝑆 ShiftStable 𝐹) = (𝑆 ShiftStable 𝐺))

Syntax hints:   → wi 4   = wceq 1562   ∩ cin 3905   ∘ ccom 5653   ShiftStable cshiftstable 38686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1565  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-rab 3417  df-in 3913  df-ss 3923  df-br 5103  df-opab 5165  df-co 5658  df-shiftstable 38986

This theorem is referenced by:  dfpet2parts2  39477  pets2eq  39481