Theorem rabeqd 38304

Description: Equality theorem for restricted class abstractions. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypothesis

Ref	Expression
rabeqd.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
rabeqd	⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜒} = {𝑥 ∈ 𝐵 ∣ 𝜒})

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝜒(𝑥)

Proof of Theorem rabeqd

Step	Hyp	Ref	Expression
1		rabeqd.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		rabeq 3166	. 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜒} = {𝑥 ∈ 𝐵 ∣ 𝜒})
3	1, 2	syl 17	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜒} = {𝑥 ∈ 𝐵 ∣ 𝜒})

Syntax hints:   → wi 4   = wceq 1475  {crab 2900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905

This theorem is referenced by:  issmflem  39613  issmfd  39621  cnfsmf  39627  issmflelem  39631  issmfgtlem  39642  issmfgt  39643  issmfled  39644  issmfgtd  39647  issmfgelem  39655