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Theorem bdbm1.3ii 9346
Description: Bounded version of bm1.3ii 3869. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bdbm1.3ii.bd BOUNDED φ
bdbm1.3ii.1 xy(φy x)
Assertion
Ref Expression
bdbm1.3ii xy(y xφ)
Distinct variable groups:   φ,x   x,y
Allowed substitution hint:   φ(y)

Proof of Theorem bdbm1.3ii
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 bdbm1.3ii.1 . . . . 5 xy(φy x)
2 elequ2 1598 . . . . . . . 8 (x = z → (y xy z))
32imbi2d 219 . . . . . . 7 (x = z → ((φy x) ↔ (φy z)))
43albidv 1702 . . . . . 6 (x = z → (y(φy x) ↔ y(φy z)))
54cbvexv 1792 . . . . 5 (xy(φy x) ↔ zy(φy z))
61, 5mpbi 133 . . . 4 zy(φy z)
7 bdbm1.3ii.bd . . . . 5 BOUNDED φ
87bdsep1 9340 . . . 4 xy(y x ↔ (y z φ))
96, 8pm3.2i 257 . . 3 (zy(φy z) xy(y x ↔ (y z φ)))
109exan 1580 . 2 z(y(φy z) xy(y x ↔ (y z φ)))
11 19.42v 1783 . . . 4 (x(y(φy z) y(y x ↔ (y z φ))) ↔ (y(φy z) xy(y x ↔ (y z φ))))
12 bimsc1 869 . . . . . 6 (((φy z) (y x ↔ (y z φ))) → (y xφ))
1312alanimi 1345 . . . . 5 ((y(φy z) y(y x ↔ (y z φ))) → y(y xφ))
1413eximi 1488 . . . 4 (x(y(φy z) y(y x ↔ (y z φ))) → xy(y xφ))
1511, 14sylbir 125 . . 3 ((y(φy z) xy(y x ↔ (y z φ))) → xy(y xφ))
1615exlimiv 1486 . 2 (z(y(φy z) xy(y x ↔ (y z φ))) → xy(y xφ))
1710, 16ax-mp 7 1 xy(y xφ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240  wex 1378  BOUNDED wbd 9267
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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-bdsep 9339
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  bj-zfpair2  9365  bj-axun2  9370  bj-uniex2  9371
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