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Theorem ralunsn 3798
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 3317 . 2 (∀𝑥 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑))
2 ralunsn.1 . . . 4 (𝑥 = 𝐵 → (𝜑𝜓))
32ralsng 3633 . . 3 (𝐵𝐶 → (∀𝑥 ∈ {𝐵}𝜑𝜓))
43anbi2d 464 . 2 (𝐵𝐶 → ((∀𝑥𝐴 𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑) ↔ (∀𝑥𝐴 𝜑𝜓)))
51, 4bitrid 192 1 (𝐵𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑥𝐴 𝜑𝜓)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wral 2455  cun 3128  {csn 3593
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 2740  df-sbc 2964  df-un 3134  df-sn 3599
This theorem is referenced by:  2ralunsn  3799  nnnninfeq2  7127
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