Theorem ifboth 2375

Description: A wff th containing a conditional operator is true when both of its cases are true.

Hypotheses

Ref	Expression
ifboth.1	|- (A = if(ph, A, B) -> (ps <-> th))
ifboth.2	|- (B = if(ph, A, B) -> (ch <-> th))

Assertion

Ref	Expression
ifboth	|- ((ps /\ ch) -> th)

Proof of Theorem ifboth

Step	Hyp	Ref	Expression
1		iftrue 2366	. . . . . 6 |- (ph -> if(ph, A, B) = A)
2	1	eqcomd 1480	. . . . 5 |- (ph -> A = if(ph, A, B))
3		ifboth.1	. . . . 5 |- (A = if(ph, A, B) -> (ps <-> th))
4	2, 3	syl 10	. . . 4 |- (ph -> (ps <-> th))
5	4	biimpa 416	. . 3 |- ((ph /\ ps) -> th)
6	5	adantrr 395	. 2 |- ((ph /\ (ps /\ ch)) -> th)
7		iffalse 2367	. . . . . 6 |- (-. ph -> if(ph, A, B) = B)
8	7	eqcomd 1480	. . . . 5 |- (-. ph -> B = if(ph, A, B))
9		ifboth.2	. . . . . 6 |- (B = if(ph, A, B) -> (ch <-> th))
10	9	bicomd 521	. . . . 5 |- (B = if(ph, A, B) -> (th <-> ch))
11	8, 10	syl 10	. . . 4 |- (-. ph -> (th <-> ch))
12	11	biimpar 417	. . 3 |- ((-. ph /\ ch) -> th)
13	12	adantrl 394	. 2 |- ((-. ph /\ (ps /\ ch)) -> th)
14	6, 13	pm2.61ian 476	1 |- ((ps /\ ch) -> th)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956  ifcif 2361

This theorem is referenced by:  ifcl 2380  keephyp 2396  xrmaxltt 5913  xrltmint 5914  maxlet 5918  lemint 5921  maxltt 5922  blin 7852  opnin 7869  xplm 7975  xpcn 7976

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-if 2362

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