Theorem hbsuc 3046

Description: Bound-variable hypothesis builder for successor.

Hypothesis

Ref	Expression
hbsuc.1	|- (y e. A -> A.x y e. A)

Assertion

Ref	Expression
hbsuc	|- (y e. suc A -> A.x y e. suc A)

Distinct variable groups:   y,A   x,y

Proof of Theorem hbsuc

Step	Hyp	Ref	Expression
1		hbsuc.1	. . 3 |- (y e. A -> A.x y e. A)
2	1	hbeleq 1570	. . 3 |- (y = A -> A.x y = A)
3	1, 2	hbor 1010	. 2 |- ((y e. A \/ y = A) -> A.x(y e. A \/ y = A))
4		visset 1816	. . 3 |- y e. V
5	4	elsuc 3044	. 2 |- (y e. suc A <-> (y e. A \/ y = A))
6	5	albii 1001	. 2 |- (A.x y e. suc A <-> A.x(y e. A \/ y = A))
7	3, 5, 6	3imtr4 219	1 |- (y e. suc A -> A.x y e. suc A)

Syntax hints:   -> wi 3   \/ wo 222  A.wal 956   = wceq 958   e. wcel 960  suc csuc 2956

This theorem is referenced by:  unblem2 4552  unblem3 4553  inf0 4615  rankid 4682

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 2053  df-sn 2416  df-pr 2417  df-suc 2960

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