Theorem rabbidva 2542

Description: Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 28-Nov-2003.)

Hypothesis

Ref	Expression
rabbidva.1	⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))

Assertion

Ref	Expression
rabbidva	⊢ (φ → {x ∈ A ∣ ψ} = {x ∈ A ∣ χ})

Distinct variable group:   φ,x

Allowed substitution hints:   ψ(x)   χ(x)   A(x)

Proof of Theorem rabbidva

Step	Hyp	Ref	Expression
1		rabbidva.1	. . 3 ⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))
2	1	ralrimiva 2386	. 2 ⊢ (φ → ∀x ∈ A (ψ ↔ χ))
3		rabbi 2481	. 2 ⊢ (∀x ∈ A (ψ ↔ χ) ↔ {x ∈ A ∣ ψ} = {x ∈ A ∣ χ})
4	2, 3	sylib 127	1 ⊢ (φ → {x ∈ A ∣ ψ} = {x ∈ A ∣ χ})

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∀wral 2300  {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-ral 2305  df-rab 2309

This theorem is referenced by:  rabbidv  2543  rabeqbidva  2547  rabbi2dva  3139  rabxfrd  4167  onsucmin  4198  seinxp  4354  fniniseg2  5232  fnniniseg2  5233  f1oresrab  5272