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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  4335  mptiniseg  5123  supeq1  6984  supeq2  6987  supeq3  6988  cardcl  7179  isnumi  7180  cardval3ex  7183  carden2bex  7187  genpdflem  7505  genipv  7507  genpelxp  7509  addcomprg  7576  mulcomprg  7578  uzval  9529  ixxval  9895  fzval  10009  hashinfom  10757  hashennn  10759  shftfn  10832  gcdval  11959  lcmval  12062  isprm  12108  odzval  12240  pceulem  12293  pceu  12294  pcval  12295  pczpre  12296  pcdiv  12301  istopon  13483  toponsspwpwg  13492  clsval  13581  neival  13613  cnpval  13668  blvalps  13858  blval  13859  limccl  14098  ellimc3apf  14099  eldvap  14121
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