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Theorem ralunsn 3559
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 3118 . 2 (x (A ∪ {B})φ ↔ (x A φ x {B}φ))
2 ralunsn.1 . . . 4 (x = B → (φψ))
32ralsng 3402 . . 3 (B 𝐶 → (x {B}φψ))
43anbi2d 437 . 2 (B 𝐶 → ((x A φ x {B}φ) ↔ (x A φ ψ)))
51, 4syl5bb 181 1 (B 𝐶 → (x (A ∪ {B})φ ↔ (x A φ ψ)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  wral 2300  cun 2909  {csn 3367
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-sbc 2759  df-un 2916  df-sn 3373
This theorem is referenced by:  2ralunsn  3560
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