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Theorem hbabd 2162
Description: Deduction form of bound-variable hypothesis builder hbab 2161.
Hypotheses
Ref Expression
hbabd.1 |- (ph -> A.xA.yph)
hbabd.2 |- (ph -> (ps -> A.xps))
Assertion
Ref Expression
hbabd |- (ph -> (z e. {y | ps} -> A.x z e. {y | ps}))
Distinct variable group:   x,z

Proof of Theorem hbabd
StepHypRef Expression
1 hbabd.1 . . . . 5 |- (ph -> A.xA.yph)
2 ax-7 1621 . . . . 5 |- (A.xA.yph -> A.yA.xph)
31, 2syl 13 . . . 4 |- (ph -> A.yA.xph)
4 hbabd.2 . . . . 5 |- (ph -> (ps -> A.xps))
542alimi 1657 . . . 4 |- (A.yA.xph -> A.yA.x(ps -> A.xps))
6 hbsb4t 1925 . . . 4 |- (A.yA.x(ps -> A.xps) -> (-. A.x x = z -> ([z / y]ps -> A.x[z / y]ps)))
73, 5, 63syl 38 . . 3 |- (ph -> (-. A.x x = z -> ([z / y]ps -> A.x[z / y]ps)))
8 ax-16 1883 . . 3 |- (A.x x = z -> ([z / y]ps -> A.x[z / y]ps))
97, 8pm2.61d2 205 . 2 |- (ph -> ([z / y]ps -> A.x[z / y]ps))
10 df-clab 2158 . 2 |- (z e. {y | ps} <-> [z / y]ps)
1110albii 1664 . 2 |- (A.x z e. {y | ps} <-> A.x[z / y]ps)
129, 10, 113imtr4g 333 1 |- (ph -> (z e. {y | ps} -> A.x z e. {y | ps}))
Colors of variables: wff set class
Syntax hints:  -. wn 2   -> wi 3  A.wal 1613   = wceq 1615   e. wcel 1617  [wsbc 1843  {cab 2157
This theorem is referenced by:  hbcsb1g 2830  hbcsbg 2832  hbifd 3229  hbiotad 5263
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1621  ax-gen 1622  ax-8 1623  ax-10 1625  ax-12 1627  ax-4 1637  ax-5o 1639  ax-6o 1642  ax-9o 1792  ax-10o 1810  ax-16 1883  ax-11o 1893
This theorem depends on definitions:  df-bi 232  df-an 435  df-ex 1645  df-sb 1845  df-clab 2158
Copyright terms: Public domain