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Theorem ralunsn 3568
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 3124 . 2 (∀𝑥 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑))
2 ralunsn.1 . . . 4 (𝑥 = 𝐵 → (𝜑𝜓))
32ralsng 3411 . . 3 (𝐵𝐶 → (∀𝑥 ∈ {𝐵}𝜑𝜓))
43anbi2d 437 . 2 (𝐵𝐶 → ((∀𝑥𝐴 𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑) ↔ (∀𝑥𝐴 𝜑𝜓)))
51, 4syl5bb 181 1 (𝐵𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑥𝐴 𝜑𝜓)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98   = wceq 1243  wcel 1393  wral 2306  cun 2915  {csn 3375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-sbc 2765  df-un 2922  df-sn 3381
This theorem is referenced by:  2ralunsn  3569
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