Theorem hblem 2186

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 1864	. 2 |- ( [ z / y ] y e. A -> A. x [ z / y ] y e. A )
3		clelsb3 2183	. 2 |- ( [ z / y ] y e. A <-> z e. A )
4	3	albii 1399	. 2 |- ( A. x [ z / y ] y e. A <-> A. x z e. A )
5	2, 3, 4	3imtr3i 198	1 |- ( z e. A -> A. x z e. A )

Syntax hints:   -> wi 4   A.wal 1282   e. wcel 1433   [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-cleq 2074  df-clel 2077

This theorem is referenced by:  nfcrii  2212