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Theorem ralunsn 3595
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 3151 . 2 (∀𝑥 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑))
2 ralunsn.1 . . . 4 (𝑥 = 𝐵 → (𝜑𝜓))
32ralsng 3438 . . 3 (𝐵𝐶 → (∀𝑥 ∈ {𝐵}𝜑𝜓))
43anbi2d 445 . 2 (𝐵𝐶 → ((∀𝑥𝐴 𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑) ↔ (∀𝑥𝐴 𝜑𝜓)))
51, 4syl5bb 185 1 (𝐵𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑥𝐴 𝜑𝜓)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  wral 2323  cun 2942  {csn 3402
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-sbc 2787  df-un 2949  df-sn 3408
This theorem is referenced by:  2ralunsn  3596
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