Theorem rabbiia 2541

Description: Equivalent wff's yield equal restricted class abstractions (inference rule). (Contributed by NM, 22-May-1999.)

Hypothesis

Ref	Expression
rabbiia.1	⊢ (x ∈ A → (φ ↔ ψ))

Assertion

Ref	Expression
rabbiia	⊢ {x ∈ A ∣ φ} = {x ∈ A ∣ ψ}

Proof of Theorem rabbiia

Step	Hyp	Ref	Expression
1		rabbiia.1	. . . 4 ⊢ (x ∈ A → (φ ↔ ψ))
2	1	pm5.32i 427	. . 3 ⊢ ((x ∈ A ∧ φ) ↔ (x ∈ A ∧ ψ))
3	2	abbii 2150	. 2 ⊢ {x ∣ (x ∈ A ∧ φ)} = {x ∣ (x ∈ A ∧ ψ)}
4		df-rab 2309	. 2 ⊢ {x ∈ A ∣ φ} = {x ∣ (x ∈ A ∧ φ)}
5		df-rab 2309	. 2 ⊢ {x ∈ A ∣ ψ} = {x ∣ (x ∈ A ∧ ψ)}
6	3, 4, 5	3eqtr4i 2067	1 ⊢ {x ∈ A ∣ φ} = {x ∈ A ∣ ψ}

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  {cab 2023  {crab 2304

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-rab 2309

This theorem is referenced by:  bm2.5ii  4188  fndmdifcom  5216  cauappcvgprlemladdru  6628  cauappcvgprlemladdrl  6629  cauappcvgpr  6634  caucvgprlemcl  6647  caucvgprlemladdrl  6649  caucvgpr  6653  ioopos  8589