Theorem hblem 2508

Description: Change the free variable of a hypothesis builder. Lemma for nfcrii 2533. (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 2159	. 2 |- ( [ z / y ] y e. A -> A. x [ z / y ] y e. A )
3		clelsb3 2506	. 2 |- ( [ z / y ] y e. A <-> z e. A )
4	3	albii 1572	. 2 |- ( A. x [ z / y ] y e. A <-> A. x z e. A )
5	2, 3, 4	3imtr3i 257	1 |- ( z e. A -> A. x z e. A )

Syntax hints:   -> wi 4   A.wal 1546   [wsb 1655   e. wcel 1721

This theorem is referenced by:  nfcrii  2533  bnj1311  29099

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385

This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-cleq 2397  df-clel 2400