Theorem bj-equsalhv 32744

Description: Version of equsalh 2294 with a dv condition, which does not require ax-13 2246. Remark: this is the same as equsalhw 2123.

Remarks: equsexvw 1932 has been moved to Main; the theorem ax13lem2 2296 has a dv version which is a simple consequence of ax5e 1841; the theorems nfeqf2 2297, dveeq2 2298, nfeqf1 2299, dveeq1 2300, nfeqf 2301, axc9 2302, ax13 2249, have dv versions which are simple consequences of ax-5 1839. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-equsalhv.nf	|- ( ps -> A. x ps )
bj-equsalhv.1	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
bj-equsalhv	|- ( A. x ( x = y -> ph ) <-> ps )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)

Proof of Theorem bj-equsalhv

Step	Hyp	Ref	Expression
1		bj-equsalhv.nf	. . 3 |- ( ps -> A. x ps )
2	1	nf5i 2024	. 2 |- F/ x ps
3		bj-equsalhv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
4	2, 3	equsalv 2108	1 |- ( A. x ( x = y -> ph ) <-> ps )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481

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-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)