Theorem shiftstableeq2 39416

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 5835	. . 3 ⊢ (𝐹 = 𝐺 → (𝑆 ∘ 𝐹) = (𝑆 ∘ 𝐺))
2		id 23	. . 3 ⊢ (𝐹 = 𝐺 → 𝐹 = 𝐺)
3	1, 2	ineq12d 4118	. 2 ⊢ (𝐹 = 𝐺 → ((𝑆 ∘ 𝐹) ∩ 𝐹) = ((𝑆 ∘ 𝐺) ∩ 𝐺))
4		df-shiftstable 39415	. 2 ⊢ (𝑆 ShiftStable 𝐹) = ((𝑆 ∘ 𝐹) ∩ 𝐹)
5		df-shiftstable 39415	. 2 ⊢ (𝑆 ShiftStable 𝐺) = ((𝑆 ∘ 𝐺) ∩ 𝐺)
6	3, 4, 5	3eqtr4g 2820	1 ⊢ (𝐹 = 𝐺 → (𝑆 ShiftStable 𝐹) = (𝑆 ShiftStable 𝐺))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∩ cin 4089   ∘ ccom 5654   ShiftStable cshiftstable 39115

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-ss 3896  df-in 4091  df-br 5103  df-opab 5167  df-co 5659  df-shiftstable 39415

This theorem is used by:  dfpet2parts2  39906  pets2eq  39910