Theorem hba1w 1707

Description: Weak version of hba1 1786. See comments for ax6w 1717. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)

Hypothesis

Ref	Expression
hbn1w.1	|- (x = y -> (ph <-> ps))

Assertion

Ref	Expression
hba1w	|- (A.xph -> A.xA.xph)

Distinct variable groups:   ph,y   ps,x   x,y

Allowed substitution hints:   ph(x)   ps(y)

Proof of Theorem hba1w

Step	Hyp	Ref	Expression
1		hbn1w.1	. . . . . . 7 |- (x = y -> (ph <-> ps))
2	1	cbvalvw 1702	. . . . . 6 |- (A.xph <-> A.yps)
3	2	a1i 10	. . . . 5 |- (x = y -> (A.xph <-> A.yps))
4	3	notbid 285	. . . 4 |- (x = y -> (-. A.xph <-> -. A.yps))
5	4	spw 1694	. . 3 |- (A.x -. A.xph -> -. A.xph)
6	5	con2i 112	. 2 |- (A.xph -> -. A.x -. A.xph)
7	4	hbn1w 1706	. 2 |- (-. A.x -. A.xph -> A.x -. A.x -. A.xph)
8	1	hbn1w 1706	. . . 4 |- (-. A.xph -> A.x -. A.xph)
9	8	con1i 121	. . 3 |- (-. A.x -. A.xph -> A.xph)
10	9	alimi 1559	. 2 |- (A.x -. A.x -. A.xph -> A.xA.xph)
11	6, 7, 10	3syl 18	1 |- (A.xph -> A.xA.xph)

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176  A.wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is referenced by: (None)