Theorem hblem 2274

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 1937	. 2 |- ( [ z / y ] y e. A -> A. x [ z / y ] y e. A )
3		clelsb1 2271	. 2 |- ( [ z / y ] y e. A <-> z e. A )
4	3	albii 1458	. 2 |- ( A. x [ z / y ] y e. A <-> A. x z e. A )
5	2, 3, 4	3imtr3i 199	1 |- ( z e. A -> A. x z e. A )

Syntax hints:   -> wi 4   A.wal 1341   [wsb 1750   e. wcel 2136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-cleq 2158  df-clel 2161

This theorem is referenced by:  nfcrii  2301