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Theorem eqabdv 2873
Description: Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.) Avoid ax-11 2155. (Revised by Wolf Lammen, 6-May-2023.)
Hypothesis
Ref Expression
eqabdv.1 (𝜑 → (𝑥𝐴𝜓))
Assertion
Ref Expression
eqabdv (𝜑𝐴 = {𝑥𝜓})
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem eqabdv
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqabdv.1 . . . 4 (𝜑 → (𝑥𝐴𝜓))
21sbbidv 2077 . . 3 (𝜑 → ([𝑦 / 𝑥]𝑥𝐴 ↔ [𝑦 / 𝑥]𝜓))
3 clelsb1 2866 . . . 4 ([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)
43bicomi 224 . . 3 (𝑦𝐴 ↔ [𝑦 / 𝑥]𝑥𝐴)
5 df-clab 2713 . . 3 (𝑦 ∈ {𝑥𝜓} ↔ [𝑦 / 𝑥]𝜓)
62, 4, 53bitr4g 314 . 2 (𝜑 → (𝑦𝐴𝑦 ∈ {𝑥𝜓}))
76eqrdv 2733 1 (𝜑𝐴 = {𝑥𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  [wsb 2062  wcel 2106  {cab 2712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814
This theorem is referenced by:  eqabcdv  2874  eqabi  2875  sbab  2887  rabeqcda  3445  iftrue  4537  iffalse  4540  dfopif  4875  iniseg  6118  setlikespec  6348  fncnvima2  7081  isoini  7358  dftpos3  8268  mapsnd  8925  hartogslem1  9580  r1val2  9875  cardval2  10029  dfac3  10159  wrdval  14552  wrdnval  14580  submgmacs  18743  submacs  18853  ablsimpgfind  20145  dfrhm2  20491  lsppr  21110  rspsn  21361  znunithash  21601  tgval3  22986  txrest  23655  xkoptsub  23678  cnextf  24090  cnblcld  24811  shft2rab  25557  sca2rab  25561  renegscl  28445  grpoinvf  30561  elpjrn  32219  ofrn2  32657  ellcsrspsn  35626  neibastop3  36345  ecres2  38261  lkrval2  39072  lshpset2N  39101  hdmapoc  41914
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