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Theorem ralsng 3632
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 3630 . 2 (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑))
2 ralsng.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
32sbcieg 2995 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
41, 3bitrd 188 1 (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  wcel 2148  wral 2455  [wsbc 2962  {csn 3592
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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  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-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-sbc 2963  df-sn 3598
This theorem is referenced by:  ralsn  3635  ralprg  3643  raltpg  3645  ralunsn  3797  iinxsng  3960  posng  4698  fimax2gtrilemstep  6899  iseqf1olemqk  10491  seq3f1olemstep  10498  fimaxre2  11230  mgm1  12783  sgrp1  12810  mnd1  12841  grp1  12970  0subg  13052  ring1  13229  2sqlem10  14392  nninfsellemdc  14679
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