Theorem 2uasban 38778

Description: Distribute the unabbreviated form of proper substitution in and out of a conjunction. (Contributed by Alan Sare, 31-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
2uasban	|- ( E. x E. y ( ( x = u /\ y = v ) /\ ( ph /\ ps ) ) <-> ( E. x E. y ( ( x = u /\ y = v ) /\ ph ) /\ E. x E. y ( ( x = u /\ y = v ) /\ ps ) ) )

Distinct variable groups:   x, u   y, u   x, v   y, v

Allowed substitution hints:   ph( x, y, v, u)   ps( x, y, v, u)

Proof of Theorem 2uasban

Step	Hyp	Ref	Expression
1		biid 251	. 2 |- ( ( E. x E. y ( ( x = u /\ y = v ) /\ ph ) /\ E. x E. y ( ( x = u /\ y = v ) /\ ps ) ) <-> ( E. x E. y ( ( x = u /\ y = v ) /\ ph ) /\ E. x E. y ( ( x = u /\ y = v ) /\ ps ) ) )
2	1	2uasbanh 38777	1 |- ( E. x E. y ( ( x = u /\ y = v ) /\ ( ph /\ ps ) ) <-> ( E. x E. y ( ( x = u /\ y = v ) /\ ph ) /\ E. x E. y ( ( x = u /\ y = v ) /\ ps ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ne 2795  df-v 3202

This theorem is referenced by: (None)