Theorem ifbothda 3693

Description: A wff th containing a conditional operator is true when both of its cases are true. (Contributed by NM, 15-Feb-2015.)

Hypotheses

Ref	Expression
ifboth.1	|- (A = if(ph, A, B) -> (ps <-> th))
ifboth.2	|- (B = if(ph, A, B) -> (ch <-> th))
ifbothda.3	|- ((et /\ ph) -> ps)
ifbothda.4	|- ((et /\ -. ph) -> ch)

Assertion

Ref	Expression
ifbothda	|- (et -> th)

Proof of Theorem ifbothda

Step	Hyp	Ref	Expression
1		ifbothda.3	. . 3 |- ((et /\ ph) -> ps)
2		iftrue 3669	. . . . . 6 |- (ph -> if(ph, A, B) = A)
3	2	eqcomd 2358	. . . . 5 |- (ph -> A = if(ph, A, B))
4		ifboth.1	. . . . 5 |- (A = if(ph, A, B) -> (ps <-> th))
5	3, 4	syl 15	. . . 4 |- (ph -> (ps <-> th))
6	5	adantl 452	. . 3 |- ((et /\ ph) -> (ps <-> th))
7	1, 6	mpbid 201	. 2 |- ((et /\ ph) -> th)
8		ifbothda.4	. . 3 |- ((et /\ -. ph) -> ch)
9		iffalse 3670	. . . . . 6 |- (-. ph -> if(ph, A, B) = B)
10	9	eqcomd 2358	. . . . 5 |- (-. ph -> B = if(ph, A, B))
11		ifboth.2	. . . . 5 |- (B = if(ph, A, B) -> (ch <-> th))
12	10, 11	syl 15	. . . 4 |- (-. ph -> (ch <-> th))
13	12	adantl 452	. . 3 |- ((et /\ -. ph) -> (ch <-> th))
14	8, 13	mpbid 201	. 2 |- ((et /\ -. ph) -> th)
15	7, 14	pm2.61dan 766	1 |- (et -> th)

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642  ifcif 3663

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  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-if 3664

This theorem is referenced by:  ifboth  3694