Theorem bj-equsal1t 32809

Description: Duplication of wl-equsal1t 33327, with shorter proof. If one imposes a DV condition on x,y , then one can use bj-alequexv 32655 and reduce axiom dependencies, and similarly for the following theorems. Note: wl-equsalcom 33328 is also interesting. (Contributed by BJ, 6-Oct-2018.)

Assertion

Ref	Expression
bj-equsal1t	|- ( F/ x ph -> ( A. x ( x = y -> ph ) <-> ph ) )

Proof of Theorem bj-equsal1t

Step	Hyp	Ref	Expression
1		bj-alequex 32708	. . 3 |- ( A. x ( x = y -> ph ) -> E. x ph )
2		19.9t 2071	. . 3 |- ( F/ x ph -> ( E. x ph <-> ph ) )
3	1, 2	syl5ib 234	. 2 |- ( F/ x ph -> ( A. x ( x = y -> ph ) -> ph ) )
4		nf5r 2064	. . 3 |- ( F/ x ph -> ( ph -> A. x ph ) )
5		ala1 1741	. . 3 |- ( A. x ph -> A. x ( x = y -> ph ) )
6	4, 5	syl6 35	. 2 |- ( F/ x ph -> ( ph -> A. x ( x = y -> ph ) ) )
7	3, 6	impbid 202	1 |- ( F/ x ph -> ( A. x ( x = y -> ph ) <-> ph ) )

Syntax hints:   -> wi 4   <-> wb 196   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  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  bj-equsal1ti  32810