Theorem rabbidv 2698

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 2697	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1332   ∈ wcel 2125  {crab 2436

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 1481  ax-11 1483  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-ral 2437  df-rab 2441

This theorem is referenced by:  rabeqbidv  2704  difeq2  3215  seex  4290  mptiniseg  5073  supeq1  6918  supeq2  6921  supeq3  6922  cardcl  7095  isnumi  7096  cardval3ex  7099  carden2bex  7103  genpdflem  7406  genipv  7408  genpelxp  7410  addcomprg  7477  mulcomprg  7479  uzval  9420  ixxval  9778  fzval  9892  hashinfom  10629  hashennn  10631  shftfn  10701  gcdval  11815  lcmval  11912  isprm  11958  istopon  12358  toponsspwpwg  12367  clsval  12458  neival  12490  cnpval  12545  blvalps  12735  blval  12736  limccl  12975  ellimc3apf  12976  eldvap  12998