Theorem hblem 2145

Description: Change the free variable of a hypothesis builder. (Contributed by NM, 5-Aug-1993.) (Revised by Andrew Salmon, 11-Jul-2011.)

Hypothesis

Ref	Expression
hblem.1	|- ( y e. A -> A. x y e. A )

Assertion

Ref	Expression
hblem	|- ( z e. A -> A. x z e. A )

Distinct variable groups:   y, A   x, z

Allowed substitution hints:   A( x, z)

Proof of Theorem hblem

Step	Hyp	Ref	Expression
1		hblem.1	. . 3 |- ( y e. A -> A. x y e. A )
2	1	hbsb 1823	. 2 |- ( [ z / y ] y e. A -> A. x [ z / y ] y e. A )
3		clelsb3 2142	. 2 |- ( [ z / y ] y e. A <-> z e. A )
4	3	albii 1359	. 2 |- ( A. x [ z / y ] y e. A <-> A. x z e. A )
5	2, 3, 4	3imtr3i 189	1 |- ( z e. A -> A. x z e. A )

Syntax hints:   -> wi 4   A.wal 1241   e. wcel 1393   [wsb 1645

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-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-cleq 2033  df-clel 2036

This theorem is referenced by:  nfcrii  2171