Theorem hbif 2373

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 2362	. 2 |- if(ph, A, B) = {z | ((z e. A /\ ph) \/ (z e. B /\ -. ph))}
2		ax-17 971	. . . . . 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 1566	. . . . 5 |- (z e. A -> A.x z e. A)
5		hbif.1	. . . . 5 |- (ph -> A.xph)
6	4, 5	hban 1009	. . . 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 1566	. . . . 5 |- (z e. B -> A.x z e. B)
9	5	hbn 1004	. . . . 5 |- (-. ph -> A.x -. ph)
10	8, 9	hban 1009	. . . 4 |- ((z e. B /\ -. ph) -> A.x(z e. B /\ -. ph))
11	6, 10	hbor 1008	. . 3 |- (((z e. A /\ ph) \/ (z e. B /\ -. ph)) -> A.x((z e. A /\ ph) \/ (z e. B /\ -. ph)))
12	11	hbab 1467	. 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 1563	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 954   e. wcel 958  {cab 1463  ifcif 2361

This theorem is referenced by:  hbrdg 3936  irredt 10322

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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