Theorem 4exbidv 1792

Description: Formula-building rule for 4 existential quantifiers (deduction rule). (Contributed by NM, 3-Aug-1995.)

Hypothesis

Ref	Expression
4exbidv.1	|- ( ph -> ( ps <-> ch ) )

Assertion

Ref	Expression
4exbidv	|- ( ph -> ( E. x E. y E. z E. w ps <-> E. x E. y E. z E. w ch ) )

Distinct variable groups:   ph, x   ph, y   ph, z   ph, w

Allowed substitution hints:   ps( x, y, z, w)   ch( x, y, z, w)

Proof of Theorem 4exbidv

Step	Hyp	Ref	Expression
1		4exbidv.1	. . 3 |- ( ph -> ( ps <-> ch ) )
2	1	2exbidv 1790	. 2 |- ( ph -> ( E. z E. w ps <-> E. z E. w ch ) )
3	2	2exbidv 1790	1 |- ( ph -> ( E. x E. y E. z E. w ps <-> E. x E. y E. z E. w ch ) )

Syntax hints:   -> wi 4   <-> wb 103   E.wex 1422

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  ceqsex8v  2645  copsex4g  4010  opbrop  4445  ovi3  5668  brecop  6262  th3q  6277  dfplpq2  6606  dfmpq2  6607  enq0sym  6684  enq0ref  6685  enq0tr  6686  enq0breq  6688  addnq0mo  6699  mulnq0mo  6700  addnnnq0  6701  mulnnnq0  6702  addsrmo  6982  mulsrmo  6983  addsrpr  6984  mulsrpr  6985