Theorem mosub 3384

Description: "At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995.)

Hypothesis

Ref	Expression
mosub.1	|- E* x ph

Assertion

Ref	Expression
mosub	|- E* x E. y ( y = A /\ ph )

Distinct variable group:   x, y, A

Allowed substitution hints:   ph( x, y)

Proof of Theorem mosub

Step	Hyp	Ref	Expression
1		moeq 3382	. 2 |- E* y y = A
2		mosub.1	. . 3 |- E* x ph
3	2	ax-gen 1722	. 2 |- A. y E* x ph
4		moexexv 2542	. 2 |- ( ( E* y y = A /\ A. y E* x ph ) -> E* x E. y ( y = A /\ ph ) )
5	1, 3, 4	mp2an 708	1 |- E* x E. y ( y = A /\ ph )

Syntax hints:   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   E*wmo 2471

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by: (None)