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Theorem sbex 1979
 Description: Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.) (Proof rewritten by Jim Kingdon, 12-Feb-2018.)
Assertion
Ref Expression
sbex
Distinct variable groups:   ,   ,
Allowed substitution hints:   (,,)

Proof of Theorem sbex
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 sbexyz 1978 . . . 4
21sbbii 1738 . . 3
3 sbexyz 1978 . . 3
42, 3bitri 183 . 2
5 ax-17 1506 . . 3
65sbco2vh 1918 . 2
7 ax-17 1506 . . . 4
87sbco2vh 1918 . . 3
98exbii 1584 . 2
104, 6, 93bitr3i 209 1
 Colors of variables: wff set class Syntax hints:   wb 104  wex 1468  wsb 1735 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736 This theorem is referenced by:  sbabel  2307  sbcex2  2962  sbcexg  2963
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