Theorem ifpfal 1389

Description: Value of the conditional operator for propositions when its first argument is false. Analogue for propositions of iffalse 3950. This is essentially dedlemb 955. (Contributed by BJ, 20-Sep-2019.)

Assertion

Ref	Expression
ifpfal

Proof of Theorem ifpfal

Step	Hyp	Ref	Expression
1		dfifp2 1382	. 2
2		simpr 461	. . . 4
3	2	com12 31	. . 3
4		ax-1 6	. . . . 5
5	4	a1i 11	. . . 4
6		pm2.21 108	. . . 4
7	5, 6	jctild 543	. . 3
8	3, 7	impbid 191	. 2
9	1, 8	syl5bb 257	1

Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  if-wif 1380

This theorem is referenced by:  ifpid  1390  bj-elimhyp  34160

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 185  df-or 370  df-an 371  df-ifp 1381