Theorem cbvral3v 1804

Description: Change bound variables of triple restricted universal quantification, using implicit substitution.

Hypotheses

Ref	Expression
cbvral3v.1	|- (x = w -> (ph <-> ch))
cbvral3v.2	|- (y = v -> (ch <-> th))
cbvral3v.3	|- (z = u -> (th <-> ps))

Assertion

Ref	Expression
cbvral3v	|- (A.x e. A A.y e. B A.z e. C ph <-> A.w e. A A.v e. B A.u e. C ps)

Distinct variable groups:   ph,w   ps,z   ch,x   ch,v   y,u,th   x,w,A   x,y,B,w   v,B   x,z,C,y,w   z,v,y,C   z,u,C

Proof of Theorem cbvral3v

Step	Hyp	Ref	Expression
1		cbvral3v.1	. . . 4 |- (x = w -> (ph <-> ch))
2	1	2ralbidv 1680	. . 3 |- (x = w -> (A.y e. B A.z e. C ph <-> A.y e. B A.z e. C ch))
3	2	cbvralv 1800	. 2 |- (A.x e. A A.y e. B A.z e. C ph <-> A.w e. A A.y e. B A.z e. C ch)
4		cbvral3v.2	. . . 4 |- (y = v -> (ch <-> th))
5		cbvral3v.3	. . . 4 |- (z = u -> (th <-> ps))
6	4, 5	cbvral2v 1803	. . 3 |- (A.y e. B A.z e. C ch <-> A.v e. B A.u e. C ps)
7	6	ralbii 1667	. 2 |- (A.w e. A A.y e. B A.z e. C ch <-> A.w e. A A.v e. B A.u e. C ps)
8	3, 7	bitr 173	1 |- (A.x e. A A.y e. B A.z e. C ph <-> A.w e. A A.v e. B A.u e. C ps)

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956  A.wral 1645

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-8 964  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-cleq 1469  df-clel 1472  df-ral 1649

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