Theorem vtocl2ga 3274

Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 20-Aug-1995.)

Hypotheses

Ref	Expression
vtocl2ga.1	|- ( x = A -> ( ph <-> ps ) )
vtocl2ga.2	|- ( y = B -> ( ps <-> ch ) )
vtocl2ga.3	|- ( ( x e. C /\ y e. D ) -> ph )

Assertion

Ref	Expression
vtocl2ga	|- ( ( A e. C /\ B e. D ) -> ch )

Distinct variable groups:   x, y, A   y, B   x, C, y   x, D, y   ps, x   ch, y

Allowed substitution hints:   ph( x, y)   ps( y)   ch( x)   B( x)

Proof of Theorem vtocl2ga

Step	Hyp	Ref	Expression
1		nfcv 2764	. 2 |- F/_ x A
2		nfcv 2764	. 2 |- F/_ y A
3		nfcv 2764	. 2 |- F/_ y B
4		nfv 1843	. 2 |- F/ x ps
5		nfv 1843	. 2 |- F/ y ch
6		vtocl2ga.1	. 2 |- ( x = A -> ( ph <-> ps ) )
7		vtocl2ga.2	. 2 |- ( y = B -> ( ps <-> ch ) )
8		vtocl2ga.3	. 2 |- ( ( x e. C /\ y e. D ) -> ph )
9	1, 2, 3, 4, 5, 6, 7, 8	vtocl2gaf 3273	1 |- ( ( A e. C /\ B e. D ) -> ch )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202

This theorem is referenced by:  solin  5058  caovcan  6838  pwfseqlem2  9481  mulcanenq  9782  ltaddnq  9796  ltrnq  9801  genpv  9821  wrdind  13476  fsumrelem  14539  imasleval  16201  fullfunc  16566  fthfunc  16567  pf1ind  19719  mretopd  20896  dvlip  23756  scvxcvx  24712  issubgoilem  28117  cnre2csqlem  29956