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Theorem sbcalf 30444
 Description: Move universal quantifier in and out of class substitution, with an explicit non-free variable condition. (Contributed by Giovanni Mascellani, 29-May-2019.)
Hypothesis
Ref Expression
sbcalf.1
Assertion
Ref Expression
sbcalf
Distinct variable group:   ,
Allowed substitution hints:   (,)   (,)

Proof of Theorem sbcalf
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 sbcalf.1 . . . 4
2 nfa1 1845 . . . 4
31, 2nfsbc 3358 . . 3
4 nfa1 1845 . . 3
53, 4nfbi 1881 . 2
6 nfv 1683 . . . . . . . 8
76sb8 2147 . . . . . . 7
87sbcbii 3396 . . . . . 6
98imbi2i 312 . . . . 5
109bicomi 202 . . . 4
1110pm5.74ri 246 . . 3
12 nfs1v 2164 . . . . . 6
131, 12nfsbc 3358 . . . . 5
14 nfv 1683 . . . . 5
15 sbequ12r 1962 . . . . . 6
1615sbcbidv 3395 . . . . 5
1713, 14, 16cbval 1994 . . . 4
1817a1i 11 . . 3
1911, 18bibi12d 321 . 2
20 sbcal 3388 . 2
215, 19, 20chvar 1982 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184  wal 1377   wceq 1379  wsb 1711  wnfc 2615  wsbc 3336 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3120  df-sbc 3337 This theorem is referenced by:  sbcalfi  30446
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