Theorem hbpr 2431

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 1570	. . 3 |- (y = A -> A.x y = A)
3		hppr.2	. . . 4 |- (y e. B -> A.x y e. B)
4	3	hbeleq 1570	. . 3 |- (y = B -> A.x y = B)
5	2, 4	hbor 1010	. 2 |- ((y = A \/ y = B) -> A.x(y = A \/ y = B))
6		visset 1816	. . 3 |- y e. V
7	6	elpr 2429	. 2 |- (y e. {A, B} <-> (y = A \/ y = B))
8	7	albii 1001	. 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 956   = wceq 958   e. wcel 960  {cpr 2415

This theorem is referenced by:  hbsn 2443  hbop 2501

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2054  df-sn 2417  df-pr 2418

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