Theorem birabi 1342

Description: Equivalent wff's yield equal restricted class abstractions (inference rule).

Hypothesis

Ref	Expression
birabi.1	⊢ (x ∈ A → (ψ ↔ χ))

Assertion

Ref	Expression
birabi	⊢ {x ∈ A∣ψ} = {x ∈ A∣χ}

Proof of Theorem birabi

Step	Hyp	Ref	Expression
1		birabi.1	. . . 4 ⊢ (x ∈ A → (ψ ↔ χ))
2	1	pm5.32i 489	. . 3 ⊢ ((x ∈ A ∧ ψ) ↔ (x ∈ A ∧ χ))
3	2	biabi 1181	. 2 ⊢ {x∣(x ∈ A ∧ ψ)} = {x∣(x ∈ A ∧ χ)}
4		df-rab 1208	. 2 ⊢ {x ∈ A∣ψ} = {x∣(x ∈ A ∧ ψ)}
5		df-rab 1208	. 2 ⊢ {x ∈ A∣χ} = {x∣(x ∈ A ∧ χ)}
6	3, 4, 5	3eqtr4 1126	1 ⊢ {x ∈ A∣ψ} = {x ∈ A∣χ}

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  {cab 1090   = wceq 1091   ∈ wcel 1092  {crab 1204

This theorem is referenced by:  bm2.5ii 2274  nlimon 2369  rankval2 3514  ranksn 3532  kmlem3 3582  hta 3619  alephsuc3 4955

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rab 1208

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