Theorem hbsbc1v 1950

Description: Bound-variable hypothesis builder for class substitution.

Hypothesis

Ref	Expression
hbsbcv.1	|- A e. V

Assertion

Ref	Expression
hbsbc1v	|- ([A / x]ph -> A.x[A / x]ph)

Distinct variable group:   x,A

Proof of Theorem hbsbc1v

Step	Hyp	Ref	Expression
1		ax-17 971	. . 3 |- (y e. A -> A.x y e. A)
2	1	hbsbc1 1949	. 2 |- ((A e. V -> [A / x]ph) -> A.x(A e. V -> [A / x]ph))
3		hbsbcv.1	. . 3 |- A e. V
4	3	a1bi 197	. 2 |- ([A / x]ph <-> (A e. V -> [A / x]ph))
5	4	albii 999	. 2 |- (A.x[A / x]ph <-> A.x(A e. V -> [A / x]ph))
6	2, 4, 5	3imtr4 219	1 |- ([A / x]ph -> A.x[A / x]ph)

Syntax hints:   -> wi 3  A.wal 954   e. wcel 958  [wsbc 1170  Vcvv 1811

This theorem is referenced by:  findes 3160  tfindes 3164  dfopab2 4113  dfoprab3 4114  nn1suc 5939  uzindOLD 6208  nn0ind-raph 6214  uzind4s 6452  fzrevralt 6519  fsum1f 7007  fsump1f 7011

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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