Theorem drnf1

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016) Usage of this theorem is discouraged because it depends on ax-13 . Use drnf1v instead. (New usage is discouraged.)

	Ref	Expression
Hypothesis	dral1.1	|- ( A. x x = y -> ( ph <-> ps ) )
Assertion	drnf1	|- ( A. x x = y -> ( F/ x ph <-> F/ y ps ) )

Proof

Step	Hyp	Ref	Expression
1		dral1.1	|- ( A. x x = y -> ( ph <-> ps ) )
2	1	dral1	|- ( A. x x = y -> ( A. x ph <-> A. y ps ) )
3	1 2	imbi12d	|- ( A. x x = y -> ( ( ph -> A. x ph ) <-> ( ps -> A. y ps ) ) )
4	3	dral1	|- ( A. x x = y -> ( A. x ( ph -> A. x ph ) <-> A. y ( ps -> A. y ps ) ) )
5		nf5	|- ( F/ x ph <-> A. x ( ph -> A. x ph ) )
6		nf5	|- ( F/ y ps <-> A. y ( ps -> A. y ps ) )
7	4 5 6	3bitr4g	|- ( A. x x = y -> ( F/ x ph <-> F/ y ps ) )