Theorem bnj1019 29543

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Assertion

Ref	Expression
bnj1019	|- ( E. p ( th /\ ch /\ ta /\ et ) <-> ( th /\ ch /\ et /\ E. p ta ) )

Distinct variable groups:   ch, p   et, p   th, p

Allowed substitution hint:   ta( p)

Proof of Theorem bnj1019

Step	Hyp	Ref	Expression
1		19.42v 1827	. 2 |- ( E. p ( ( th /\ ch /\ et ) /\ ta ) <-> ( ( th /\ ch /\ et ) /\ E. p ta ) )
2		bnj258 29465	. . 3 |- ( ( th /\ ch /\ ta /\ et ) <-> ( ( th /\ ch /\ et ) /\ ta ) )
3	2	exbii 1712	. 2 |- ( E. p ( th /\ ch /\ ta /\ et ) <-> E. p ( ( th /\ ch /\ et ) /\ ta ) )
4		df-bnj17 29444	. 2 |- ( ( th /\ ch /\ et /\ E. p ta ) <-> ( ( th /\ ch /\ et ) /\ E. p ta ) )
5	1, 3, 4	3bitr4i 280	1 |- ( E. p ( th /\ ch /\ ta /\ et ) <-> ( th /\ ch /\ et /\ E. p ta ) )

Syntax hints:   <-> wb 187   /\ wa 370   /\ w3a 982   E.wex 1657   /\ w-bnj17 29443

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798

This theorem depends on definitions:  df-bi 188  df-an 372  df-3an 984  df-ex 1658  df-bnj17 29444

This theorem is referenced by:  bnj1018  29725  bnj1020  29726  bnj1021  29727