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Theorem ralunsn 3879
 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 (B C → (x (A ∪ {B})φ ↔ (x A φ ψ)))
Distinct variable groups:   x,B   ψ,x
Allowed substitution hints:   φ(x)   A(x)   C(x)

Proof of Theorem ralunsn
StepHypRef Expression
1 ralunb 3444 . 2 (x (A ∪ {B})φ ↔ (x A φ x {B}φ))
2 ralunsn.1 . . . 4 (x = B → (φψ))
32ralsng 3765 . . 3 (B C → (x {B}φψ))
43anbi2d 684 . 2 (B C → ((x A φ x {B}φ) ↔ (x A φ ψ)))
51, 4syl5bb 248 1 (B C → (x (A ∪ {B})φ ↔ (x A φ ψ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2614   ∪ cun 3207  {csn 3737 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741 This theorem is referenced by:  2ralunsn  3880
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