Theorem hbsbc1g 1955

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 1950.)

Hypothesis

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

Assertion

Ref	Expression
hbsbc1g	|- (A e. B -> ([A / x]ph -> A.x[A / x]ph))

Distinct variable groups:   x,y   y,A

Proof of Theorem hbsbc1g

Step	Hyp	Ref	Expression
1		dfsbcq 1950	. . 3 |- (z = A -> ([z / x]ph <-> [A / x]ph))
2		ax-17 975	. . . . . 6 |- (y e. z -> A.x y e. z)
3		hbsbc1g.1	. . . . . 6 |- (y e. A -> A.x y e. A)
4	2, 3	hbeq 1572	. . . . 5 |- (z = A -> A.x z = A)
5	4, 1	albid 1110	. . . 4 |- (z = A -> (A.x[z / x]ph <-> A.x[A / x]ph))
6		hbs1 1338	. . . 4 |- ([z / x]ph -> A.x[z / x]ph)
7	5, 6	syl5bi 208	. . 3 |- (z = A -> ([z / x]ph -> A.x[A / x]ph))
8	1, 7	sylbird 205	. 2 |- (z = A -> ([A / x]ph -> A.x[A / x]ph))
9	8	vtocleg 1862	1 |- (A e. B -> ([A / x]ph -> A.x[A / x]ph))

Syntax hints:   -> wi 3  A.wal 958   = wceq 960   e. wcel 962  [wsbc 1176

This theorem is referenced by:  hbsbc1 1956  hbsbc1gd 1991  hbcsb1g 2033  sbcbrg 2675  reuuniss 2903  reuuniss2 2905

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-10 970  ax-12 972  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1129  ax-10o 1146  ax-16 1216  ax-11o 1224  ax-ext 1466

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 985  df-sb 1178  df-clab 1471  df-cleq 1476  df-clel 1479  df-v 1819  df-sbc 1949

Copyright terms: Public domain