Theorem shiftstableeq2 39606

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 5841	. . 3 ⊢ (𝐹 = 𝐺 → (𝑆 ∘ 𝐹) = (𝑆 ∘ 𝐺))
2		id 23	. . 3 ⊢ (𝐹 = 𝐺 → 𝐹 = 𝐺)
3	1, 2	ineq12d 4121	. 2 ⊢ (𝐹 = 𝐺 → ((𝑆 ∘ 𝐹) ∩ 𝐹) = ((𝑆 ∘ 𝐺) ∩ 𝐺))
4		df-shiftstable 39605	. 2 ⊢ (𝑆 ShiftStable 𝐹) = ((𝑆 ∘ 𝐹) ∩ 𝐹)
5		df-shiftstable 39605	. 2 ⊢ (𝑆 ShiftStable 𝐺) = ((𝑆 ∘ 𝐺) ∩ 𝐺)
6	3, 4, 5	3eqtr4g 2823	1 ⊢ (𝐹 = 𝐺 → (𝑆 ShiftStable 𝐹) = (𝑆 ShiftStable 𝐺))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∩ cin 4092   ∘ ccom 5659   ShiftStable cshiftstable 39305

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-rab 3417  df-ss 3901  df-in 4094  df-br 5107  df-opab 5171  df-co 5664  df-shiftstable 39605

This theorem is used by:  dfpet2parts2  40096  pets2eq  40100