Theorem hblem 2731

Description: Change the free variable of a hypothesis builder. Lemma for nfcrii 2757. (Contributed by NM, 21-Jun-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 2441	. 2 |- ( [ z / y ] y e. A -> A. x [ z / y ] y e. A )
3		clelsb3 2729	. 2 |- ( [ z / y ] y e. A <-> z e. A )
4	3	albii 1747	. 2 |- ( A. x [ z / y ] y e. A <-> A. x z e. A )
5	2, 3, 4	3imtr3i 280	1 |- ( z e. A -> A. x z e. A )

Syntax hints:   -> wi 4   A.wal 1481   [wsb 1880   e. wcel 1990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-cleq 2615  df-clel 2618

This theorem is referenced by:  nfcrii  2757  bnj1311  31092