Theorem hbsbc1 1949

Description: Bound-variable hypothesis builder for class substitution. (The antecedent ensures that A is a set, which is necessary if we restrict ourselves to using only the "weak" class substitution definition dfsbcq 1943.)

Hypothesis

Ref	Expression
hbsbc1g.1	|- (y e. A -> A.x y e. A)

Assertion

Ref	Expression
hbsbc1	|- ((A e. B -> [A / x]ph) -> A.x(A e. B -> [A / x]ph))

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

Proof of Theorem hbsbc1

Step	Hyp	Ref	Expression
1		hbsbc1g.1	. . 3 |- (y e. A -> A.x y e. A)
2		ax-17 971	. . 3 |- (y e. B -> A.x y e. B)
3	1, 2	hbel 1566	. 2 |- (A e. B -> A.x A e. B)
4	1	hbsbc1g 1948	. 2 |- (A e. B -> ([A / x]ph -> A.x[A / x]ph))
5	3, 4	hbim1 1103	1 |- ((A e. B -> [A / x]ph) -> A.x(A e. B -> [A / x]ph))

Syntax hints:   -> wi 3  A.wal 954   e. wcel 958  [wsbc 1170

This theorem is referenced by:  hbsbc1v 1950  sbc5g 1954  sbc6g 1955  elrabsf 1963  sbcel1gv 1980  sbcel2gv 1981  reuuni4 2887  nn1suc 5939

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-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-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-sbc 1942

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