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Theorem sbcexf 28864
 Description: Move existential quantifier in and out of class substitution, with an explicit non-free variable condition. (Contributed by Giovanni Mascellani, 29-May-2019.)
Hypothesis
Ref Expression
sbcexf.1
Assertion
Ref Expression
sbcexf
Distinct variable group:   ,
Allowed substitution hints:   (,)   (,)

Proof of Theorem sbcexf
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 sbcexf.1 . . . 4
2 nfe1 1778 . . . 4
31, 2nfsbc 3201 . . 3
4 nfe1 1778 . . 3
53, 4nfbi 1866 . 2
6 nfv 1673 . . . . . . . 8
76sb8e 2125 . . . . . . 7
87sbcbii 3239 . . . . . 6
98imbi2i 312 . . . . 5
109bicomi 202 . . . 4
1110pm5.74ri 246 . . 3
12 nfs1v 2142 . . . . . 6
131, 12nfsbc 3201 . . . . 5
14 nfv 1673 . . . . 5
15 sbequ12r 1937 . . . . . 6
1615sbcbidv 3238 . . . . 5
1713, 14, 16cbvex 1970 . . . 4
1817a1i 11 . . 3
1911, 18bibi12d 321 . 2
20 sbcex2 3233 . 2
215, 19, 20chvar 1957 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wceq 1369  wex 1586  wsb 1700  wnfc 2560  wsbc 3179 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2418 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-v 2968  df-sbc 3180 This theorem is referenced by:  sbcexfi  28866
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