Theorem rabbidv 2678

Description: Equivalent wff's yield equal restricted class abstractions (deduction form). (Contributed by NM, 10-Feb-1995.)

Hypothesis

Ref	Expression
rabbidv.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

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

Distinct variable group:   𝜑,𝑥

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

Proof of Theorem rabbidv

Step	Hyp	Ref	Expression
1		rabbidv.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	adantr 274	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))
3	2	rabbidva 2677	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1332   ∈ wcel 1481  {crab 2421

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-ral 2422  df-rab 2426

This theorem is referenced by:  rabeqbidv  2684  difeq2  3193  seex  4265  mptiniseg  5041  supeq1  6881  supeq2  6884  supeq3  6885  cardcl  7054  isnumi  7055  cardval3ex  7058  carden2bex  7062  genpdflem  7339  genipv  7341  genpelxp  7343  addcomprg  7410  mulcomprg  7412  uzval  9352  ixxval  9709  fzval  9823  hashinfom  10556  hashennn  10558  shftfn  10628  gcdval  11684  lcmval  11780  isprm  11826  istopon  12219  toponsspwpwg  12228  clsval  12319  neival  12351  cnpval  12406  blvalps  12596  blval  12597  limccl  12836  ellimc3apf  12837  eldvap  12859