Theorem ceqsalgALT 3231

Description: Alternate proof of ceqsalg 3230, not using ceqsalt 3228. (Contributed by NM, 29-Oct-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by BJ, 29-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ceqsalg.1	|- F/ x ps
ceqsalg.2	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
ceqsalgALT	|- ( A e. V -> ( A. x ( x = A -> ph ) <-> ps ) )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   ps( x)   V( x)

Proof of Theorem ceqsalgALT

Step	Hyp	Ref	Expression
1		elisset 3215	. . 3 |- ( A e. V -> E. x x = A )
2		nfa1 2028	. . . 4 |- F/ x A. x ( x = A -> ph )
3		ceqsalg.1	. . . 4 |- F/ x ps
4		ceqsalg.2	. . . . . . 7 |- ( x = A -> ( ph <-> ps ) )
5	4	biimpd 219	. . . . . 6 |- ( x = A -> ( ph -> ps ) )
6	5	a2i 14	. . . . 5 |- ( ( x = A -> ph ) -> ( x = A -> ps ) )
7	6	sps 2055	. . . 4 |- ( A. x ( x = A -> ph ) -> ( x = A -> ps ) )
8	2, 3, 7	exlimd 2087	. . 3 |- ( A. x ( x = A -> ph ) -> ( E. x x = A -> ps ) )
9	1, 8	syl5com 31	. 2 |- ( A e. V -> ( A. x ( x = A -> ph ) -> ps ) )
10	4	biimprcd 240	. . 3 |- ( ps -> ( x = A -> ph ) )
11	3, 10	alrimi 2082	. 2 |- ( ps -> A. x ( x = A -> ph ) )
12	9, 11	impbid1 215	1 |- ( A e. V -> ( A. x ( x = A -> ph ) <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   E.wex 1704   F/wnf 1708   e. wcel 1990

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-12 2047  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-v 3202

This theorem is referenced by: (None)