Theorem 4exbidv 1842

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 1840	. 2 |- ( ph -> ( E. z E. w ps <-> E. z E. w ch ) )
3	2	2exbidv 1840	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 104   E.wex 1468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  ceqsex8v  2731  copsex4g  4169  opbrop  4618  ovi3  5907  brecop  6519  th3q  6534  dfplpq2  7162  dfmpq2  7163  enq0sym  7240  enq0ref  7241  enq0tr  7242  enq0breq  7244  addnq0mo  7255  mulnq0mo  7256  addnnnq0  7257  mulnnnq0  7258  addsrmo  7551  mulsrmo  7552  addsrpr  7553  mulsrpr  7554