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Theorem rabbidv 2726
Description: Equivalent wff's yield equal restricted class abstractions (deduction form). (Contributed by NM, 10-Feb-1995.)
Hypothesis
Ref Expression
rabbidv.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
rabbidv (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem rabbidv
StepHypRef Expression
1 rabbidv.1 . . 3 (𝜑 → (𝜓𝜒))
21adantr 276 . 2 ((𝜑𝑥𝐴) → (𝜓𝜒))
32rabbidva 2725 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  wcel 2148  {crab 2459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-ral 2460  df-rab 2464
This theorem is referenced by:  rabeqbidv  2732  difeq2  3247  seex  4331  mptiniseg  5118  supeq1  6978  supeq2  6981  supeq3  6982  cardcl  7173  isnumi  7174  cardval3ex  7177  carden2bex  7181  genpdflem  7484  genipv  7486  genpelxp  7488  addcomprg  7555  mulcomprg  7557  uzval  9506  ixxval  9870  fzval  9984  hashinfom  10729  hashennn  10731  shftfn  10804  gcdval  11930  lcmval  12033  isprm  12079  odzval  12211  pceulem  12264  pceu  12265  pcval  12266  pczpre  12267  pcdiv  12272  istopon  13144  toponsspwpwg  13153  clsval  13244  neival  13276  cnpval  13331  blvalps  13521  blval  13522  limccl  13761  ellimc3apf  13762  eldvap  13784
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