Theorem hbpr 2423

Description: Bound-variable hypothesis builder for unordered pairs.

Hypotheses

Ref	Expression
hbpr.1	|- (y e. A -> A.x y e. A)
hppr.2	|- (y e. B -> A.x y e. B)

Assertion

Ref	Expression
hbpr	|- (y e. {A, B} -> A.x y e. {A, B})

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

Proof of Theorem hbpr

Step	Hyp	Ref	Expression
1		hbpr.1	. . . 4 |- (y e. A -> A.x y e. A)
2	1	hbeleq 1565	. . 3 |- (y = A -> A.x y = A)
3		hppr.2	. . . 4 |- (y e. B -> A.x y e. B)
4	3	hbeleq 1565	. . 3 |- (y = B -> A.x y = B)
5	2, 4	hbor 1007	. 2 |- ((y = A \/ y = B) -> A.x(y = A \/ y = B))
6		visset 1810	. . 3 |- y e. V
7	6	elpr 2421	. 2 |- (y e. {A, B} <-> (y = A \/ y = B))
8	7	albii 998	. 2 |- (A.x y e. {A, B} <-> A.x(y = A \/ y = B))
9	5, 7, 8	3imtr4 219	1 |- (y e. {A, B} -> A.x y e. {A, B})

Syntax hints:   -> wi 3   \/ wo 222  A.wal 953   = wceq 955   e. wcel 957  {cpr 2407

This theorem is referenced by:  hbsn 2435  hbop 2493

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809  df-un 2047  df-sn 2409  df-pr 2410

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