Theorem hbif 2363

Description: Bound-variable hypothesis builder for a conditional operator.

Hypotheses

Ref	Expression
hbif.1	|- (ph -> A.xph)
hbif.2	|- (y e. A -> A.x y e. A)
hbif.3	|- (y e. B -> A.x y e. B)

Assertion

Ref	Expression
hbif	|- (y e. if(ph, A, B) -> A.x y e. if(ph, A, B))

Distinct variable groups:   x,y   y,A   y,B

Proof of Theorem hbif

Step	Hyp	Ref	Expression
1		df-if 2352	. 2 |- if(ph, A, B) = {z | ((z e. A /\ ph) \/ (z e. B /\ -. ph))}
2		ax-17 968	. . . . . 6 |- (y e. z -> A.x y e. z)
3		hbif.2	. . . . . 6 |- (y e. A -> A.x y e. A)
4	2, 3	hbel 1558	. . . . 5 |- (z e. A -> A.x z e. A)
5		hbif.1	. . . . 5 |- (ph -> A.xph)
6	4, 5	hban 1006	. . . 4 |- ((z e. A /\ ph) -> A.x(z e. A /\ ph))
7		hbif.3	. . . . . 6 |- (y e. B -> A.x y e. B)
8	2, 7	hbel 1558	. . . . 5 |- (z e. B -> A.x z e. B)
9	5	hbn 1001	. . . . 5 |- (-. ph -> A.x -. ph)
10	8, 9	hban 1006	. . . 4 |- ((z e. B /\ -. ph) -> A.x(z e. B /\ -. ph))
11	6, 10	hbor 1005	. . 3 |- (((z e. A /\ ph) \/ (z e. B /\ -. ph)) -> A.x((z e. A /\ ph) \/ (z e. B /\ -. ph)))
12	11	hbab 1460	. 2 |- (y e. {z | ((z e. A /\ ph) \/ (z e. B /\ -. ph))} -> A.x y e. {z | ((z e. A /\ ph) \/ (z e. B /\ -. ph))})
13	1, 12	hbxfr 1555	1 |- (y e. if(ph, A, B) -> A.x y e. if(ph, A, B))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 222   /\ wa 223  A.wal 951   e. wcel 955  {cab 1456  ifcif 2351

This theorem is referenced by:  hbrdg 3921  irredt 10230

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-if 2352

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