Theorem wl-lem-moexsb 31604

Description: The antecedent A. x ( ph -> x = z ) relates to E* x ph, but is better suited for usage in proofs. Note that no distinct variable restriction is placed on ph.

This theorem provides a basic working step in proving theorems about E* or E!. (Contributed by Wolf Lammen, 3-Oct-2019.)

Assertion

Ref	Expression
wl-lem-moexsb	|- ( A. x ( ph -> x = z ) -> ( E. x ph <-> [ z / x ] ph ) )

Distinct variable group:   x, z

Allowed substitution hints:   ph( x, z)

Proof of Theorem wl-lem-moexsb

Step	Hyp	Ref	Expression
1		nfa1 1954	. . 3 |- F/ x A. x ( ph -> x = z )
2		nfs1v 2233	. . 3 |- F/ x [ z / x ] ph
3		sp 1912	. . . . 5 |- ( A. x ( ph -> x = z ) -> ( ph -> x = z ) )
4		ax12v 1908	. . . . 5 |- ( x = z -> ( ph -> A. x ( x = z -> ph ) ) )
5	3, 4	syli 38	. . . 4 |- ( A. x ( ph -> x = z ) -> ( ph -> A. x ( x = z -> ph ) ) )
6		sb2 2147	. . . 4 |- ( A. x ( x = z -> ph ) -> [ z / x ] ph )
7	5, 6	syl6 34	. . 3 |- ( A. x ( ph -> x = z ) -> ( ph -> [ z / x ] ph ) )
8	1, 2, 7	exlimd 1972	. 2 |- ( A. x ( ph -> x = z ) -> ( E. x ph -> [ z / x ] ph ) )
9		spsbe 1793	. 2 |- ( [ z / x ] ph -> E. x ph )
10	8, 9	impbid1 206	1 |- ( A. x ( ph -> x = z ) -> ( E. x ph <-> [ z / x ] ph ) )

Syntax hints:   -> wi 4   <-> wb 187   A.wal 1435   E.wex 1659   [wsb 1789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-12 1907  ax-13 2055

This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664  df-sb 1790

This theorem is referenced by: (None)