Theorem 4exbidv 1923

Description: Formula-building rule for 4 existential quantifiers (deduction form). (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 1921	. 2 |- ( ph -> ( E. z E. w ps <-> E. z E. w ch ) )
3	2	2exbidv 1921	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 105   E.wex 1545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  ceqsex8v  2868  copsex4g  4382  opbrop  4849  ovi3  6216  brecop  6889  th3q  6904  dfplpq2  7711  dfmpq2  7712  enq0sym  7789  enq0ref  7790  enq0tr  7791  enq0breq  7793  addnq0mo  7804  mulnq0mo  7805  addnnnq0  7806  mulnnnq0  7807  addsrmo  8100  mulsrmo  8101  addsrpr  8102  mulsrpr  8103