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Theorem ralsng 3599
 Description: Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
Hypothesis
Ref Expression
ralsng.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
ralsng (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑𝜓))
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem ralsng
StepHypRef Expression
1 ralsns 3597 . 2 (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑))
2 ralsng.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
32sbcieg 2969 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
41, 3bitrd 187 1 (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   = wceq 1335   ∈ wcel 2128  ∀wral 2435  [wsbc 2937  {csn 3560 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-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-sbc 2938  df-sn 3566 This theorem is referenced by:  ralsn  3602  ralprg  3610  raltpg  3612  ralunsn  3760  iinxsng  3922  posng  4655  fimax2gtrilemstep  6838  iseqf1olemqk  10375  seq3f1olemstep  10382  fimaxre2  11108  nninfsellemdc  13544
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