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Theorem ralunsn 4392
Description: Restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.) (Revised by Mario Carneiro, 23-Apr-2015.)
Hypothesis
Ref Expression
ralunsn.1 (𝑥 = 𝐵 → (𝜑𝜓))
Assertion
Ref Expression
ralunsn (𝐵𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑥𝐴 𝜑𝜓)))
Distinct variable groups:   𝑥,𝐵   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐶(𝑥)

Proof of Theorem ralunsn
StepHypRef Expression
1 ralunb 3774 . 2 (∀𝑥 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑))
2 ralunsn.1 . . . 4 (𝑥 = 𝐵 → (𝜑𝜓))
32ralsng 4191 . . 3 (𝐵𝐶 → (∀𝑥 ∈ {𝐵}𝜑𝜓))
43anbi2d 739 . 2 (𝐵𝐶 → ((∀𝑥𝐴 𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑) ↔ (∀𝑥𝐴 𝜑𝜓)))
51, 4syl5bb 272 1 (𝐵𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑥𝐴 𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  cun 3554  {csn 4150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-v 3188  df-sbc 3419  df-un 3561  df-sn 4151
This theorem is referenced by:  2ralunsn  4393  symgextfo  17766  gsmsymgrfixlem1  17771  gsmsymgreqlem2  17775  symgfixf1  17781  cply1coe0bi  19592  scmatf1  20259  mdetunilem9  20348  m2cpminvid2lem  20481  tgcgr4  25333  clwlkclwwlklem2a1  26767  clwlksf1clwwlklem  26841  disjunsn  29264
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