Theorem bj-dral1v 32748

Description: Version of dral1 2325 with a dv condition, which does not require ax-13 2246. Remark: the corresponding versions for dral2 2324 and drex2 2328 are instances of albidv 1849 and exbidv 1850 respectively. (Contributed by BJ, 17-Jun-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-dral1v.1	|- ( A. x x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
bj-dral1v	|- ( A. x x = y -> ( A. x ph <-> A. y ps ) )

Distinct variable group:   x, y

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

Proof of Theorem bj-dral1v

Step	Hyp	Ref	Expression
1		nfa1 2028	. . 3 |- F/ x A. x x = y
2		bj-dral1v.1	. . 3 |- ( A. x x = y -> ( ph <-> ps ) )
3	1, 2	albid 2090	. 2 |- ( A. x x = y -> ( A. x ph <-> A. x ps ) )
4		bj-axc11v 32747	. . 3 |- ( A. x x = y -> ( A. x ps -> A. y ps ) )
5		axc11r 2187	. . 3 |- ( A. x x = y -> ( A. y ps -> A. x ps ) )
6	4, 5	impbid 202	. 2 |- ( A. x x = y -> ( A. x ps <-> A. y ps ) )
7	3, 6	bitrd 268	1 |- ( A. x x = y -> ( A. x ph <-> A. y 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-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  bj-drex1v  32749  bj-drnf1v  32750