Theorem rabeqf 1855

Description: Equality theorem for restricted class abstractions, with bound-variable hypotheses instead of distinct variable restrictions.

Hypotheses

Ref	Expression
rabeqf.1	⊢ (y ∈ A → ∀x y ∈ A)
rabeqf.2	⊢ (y ∈ B → ∀x y ∈ B)

Assertion

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

Distinct variable groups:   y,A   y,B   x,y

Proof of Theorem rabeqf

Step	Hyp	Ref	Expression
1		rabeqf.1	. . . 4 ⊢ (y ∈ A → ∀x y ∈ A)
2		rabeqf.2	. . . 4 ⊢ (y ∈ B → ∀x y ∈ B)
3	1, 2	hbeq 1612	. . 3 ⊢ (A = B → ∀x A = B)
4		eleq2 1582	. . . 4 ⊢ (A = B → (x ∈ A ↔ x ∈ B))
5	4	anbi1d 628	. . 3 ⊢ (A = B → ((x ∈ A ⋀ φ) ↔ (x ∈ B ⋀ φ)))
6	3, 5	abbid 1623	. 2 ⊢ (A = B → {x∣(x ∈ A ⋀ φ)} = {x∣(x ∈ B ⋀ φ)})
7		df-rab 1699	. 2 ⊢ {x ∈ A∣φ} = {x∣(x ∈ A ⋀ φ)}
8		df-rab 1699	. 2 ⊢ {x ∈ B∣φ} = {x∣(x ∈ B ⋀ φ)}
9	6, 7, 8	3eqtr4g 1578	1 ⊢ (A = B → {x ∈ A∣φ} = {x ∈ B∣φ})

Syntax hints:   → wi 3   ⋀ wa 230  ∀wal 995   = wceq 997   ∈ wcel 999  {cab 1509  {crab 1695

This theorem is referenced by:  rabeq 1856  hta 4790

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-12 1009  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504

This theorem depends on definitions:  df-bi 154  df-an 232  df-ex 1022  df-sb 1214  df-clab 1510  df-cleq 1515  df-clel 1518  df-rab 1699

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