Theorem 19.9d 2070

Description: A deduction version of one direction of 19.9 2072. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) Revised to shorten other proofs. (Revised by Wolf Lammen, 14-Jul-2020.) df-nf 1710 changed. (Revised by Wolf Lammen, 11-Sep-2021.)

Hypothesis

Ref	Expression
19.9d.1	|- ( ps -> F/ x ph )

Assertion

Ref	Expression
19.9d	|- ( ps -> ( E. x ph -> ph ) )

Proof of Theorem 19.9d

Step	Hyp	Ref	Expression
1		19.9d.1	. . 3 |- ( ps -> F/ x ph )
2		df-nf 1710	. . 3 |- ( F/ x ph <-> ( E. x ph -> A. x ph ) )
3	1, 2	sylib 208	. 2 |- ( ps -> ( E. x ph -> A. x ph ) )
4		sp 2053	. 2 |- ( A. x ph -> ph )
5	3, 4	syl6 35	1 |- ( ps -> ( E. x ph -> ph ) )

Syntax hints:   -> wi 4   A.wal 1481   E.wex 1704   F/wnf 1708

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-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-nf 1710

This theorem is referenced by:  19.9t  2071  exdistrf  2333  equvel  2347  copsexg  4956  19.9d2rf  29318  wl-exeq  33321  spd  42425