Theorem uunT1 39007

Description: A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 3-Dec-2015.) Proof was revised to accomodate a possible future version of df-tru 1486. (Revised by David A. Wheeler, 8-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
uunT1.1	|- ( ( T. /\ ph ) -> ps )

Assertion

Ref	Expression
uunT1	|- ( ph -> ps )

Proof of Theorem uunT1

Step	Hyp	Ref	Expression
1		orc 400	. . 3 |- ( ph -> ( ph \/ -. ph ) )
2		tru 1487	. . . . 5 |- T.
3		biid 251	. . . . 5 |- ( ph <-> ph )
4	2, 3	2th 254	. . . 4 |- ( T. <-> ( ph <-> ph ) )
5		exmid 431	. . . . . 6 |- ( ph \/ -. ph )
6	5	a1i 11	. . . . 5 |- ( ( ph <-> ph ) -> ( ph \/ -. ph ) )
7		biidd 252	. . . . 5 |- ( ( ph \/ -. ph ) -> ( ph <-> ph ) )
8	6, 7	impbii 199	. . . 4 |- ( ( ph <-> ph ) <-> ( ph \/ -. ph ) )
9	4, 8	bitri 264	. . 3 |- ( T. <-> ( ph \/ -. ph ) )
10	1, 9	sylibr 224	. 2 |- ( ph -> T. )
11		uunT1.1	. 2 |- ( ( T. /\ ph ) -> ps )
12	10, 11	mpancom 703	1 |- ( ph -> ps )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   T. wtru 1484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486

This theorem is referenced by:  uunT21  39009  sspwimpALT  39161