Theorem 2rexbidv 1673

Description: Formula-building rule for restricted existential quantifiers (deduction rule).

Hypothesis

Ref	Expression
2ralbidv.1	|- (ph -> (ps <-> ch))

Assertion

Ref	Expression
2rexbidv	|- (ph -> (E.x e. A E.y e. B ps <-> E.x e. A E.y e. B ch))

Distinct variable groups:   ph,x   ph,y

Proof of Theorem 2rexbidv

Step	Hyp	Ref	Expression
1		2ralbidv.1	. . 3 |- (ph -> (ps <-> ch))
2	1	rexbidv 1656	. 2 |- (ph -> (E.y e. B ps <-> E.y e. B ch))
3	2	rexbidv 1656	1 |- (ph -> (E.x e. A E.y e. B ps <-> E.x e. A E.y e. B ch))

Syntax hints:   -> wi 3   <-> wb 146  E.wrex 1638

This theorem is referenced by:  f1oiso 3889  oprvalelrn 4024  brdom7disj 4776  brdom6disj 4777  axcnre 5258  elq 6195  hausnei 7723  pjtht 9149  shselt 9193  iseuctopg 10389

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-17 968  ax-4 970  ax-5o 972

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-rex 1642

Copyright terms: Public domain