Theorem hblem 2196

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 1872	. 2 |- ( [ z / y ] y e. A -> A. x [ z / y ] y e. A )
3		clelsb3 2193	. 2 |- ( [ z / y ] y e. A <-> z e. A )
4	3	albii 1405	. 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 1288   e. wcel 1439   [wsb 1693

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-cleq 2082  df-clel 2085

This theorem is referenced by:  nfcrii  2222