Theorem hbabd 1466

Description: Deduction form of bound-variable hypothesis builder hbab 1465.

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

Step	Hyp	Ref	Expression
1		hbabd.1	. . . . 5 |- (ph -> A.xA.yph)
2		ax-7 960	. . . . 5 |- (A.xA.yph -> A.yA.xph)
3	1, 2	syl 10	. . . 4 |- (ph -> A.yA.xph)
4		hbabd.2	. . . . 5 |- (ph -> (ps -> A.xps))
5	4	19.20i2 991	. . . 4 |- (A.yA.xph -> A.yA.x(ps -> A.xps))
6		hbsb4t 1247	. . . 4 |- (A.yA.x(ps -> A.xps) -> (-. A.x x = z -> ([z / y]ps -> A.x[z / y]ps)))
7	3, 5, 6	3syl 20	. . 3 |- (ph -> (-. A.x x = z -> ([z / y]ps -> A.x[z / y]ps)))
8		ax-16 1208	. . 3 |- (A.x x = z -> ([z / y]ps -> A.x[z / y]ps))
9	7, 8	pm2.61d2 129	. 2 |- (ph -> ([z / y]ps -> A.x[z / y]ps))
10		df-clab 1462	. 2 |- (z e. {y | ps} <-> [z / y]ps)
11	10	albii 997	. 2 |- (A.x z e. {y | ps} <-> A.x[z / y]ps)
12	9, 10, 11	3imtr4g 552	1 |- (ph -> (z e. {y | ps} -> A.x z e. {y | ps}))

Syntax hints:  -. wn 2   -> wi 3  A.wal 952   = wceq 954   e. wcel 956  [wsbc 1168  {cab 1461

This theorem is referenced by:  hbcsb1g 2020  hbcsbg 2022

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462

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