Theorem wl-dral1d

Description: A version of dral1 with a context. Note: At first glance one might be tempted to generalize this (or a similar) theorem by weakening the first two hypotheses adding a x = y , A. x x = y or ph antecedent. wl-equsal1i and nf5di show that this is in fact pointless. (Contributed by Wolf Lammen, 28-Jul-2019)

	Ref	Expression
Hypotheses	wl-dral1d.1	|- F/ x ph
	wl-dral1d.2	|- F/ y ph
	wl-dral1d.3	|- ( ph -> ( x = y -> ( ps <-> ch ) ) )
Assertion	wl-dral1d	|- ( ph -> ( A. x x = y -> ( A. x ps <-> A. y ch ) ) )

Proof

Step	Hyp	Ref	Expression
1		wl-dral1d.1	|- F/ x ph
2		wl-dral1d.2	|- F/ y ph
3		wl-dral1d.3	|- ( ph -> ( x = y -> ( ps <-> ch ) ) )
4	3	com12	|- ( x = y -> ( ph -> ( ps <-> ch ) ) )
5	4	pm5.74d	|- ( x = y -> ( ( ph -> ps ) <-> ( ph -> ch ) ) )
6	5	sps	|- ( A. x x = y -> ( ( ph -> ps ) <-> ( ph -> ch ) ) )
7	6	dral1	|- ( A. x x = y -> ( A. x ( ph -> ps ) <-> A. y ( ph -> ch ) ) )
8	1	19.21	|- ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) )
9	2	19.21	|- ( A. y ( ph -> ch ) <-> ( ph -> A. y ch ) )
10	7 8 9	3bitr3g	|- ( A. x x = y -> ( ( ph -> A. x ps ) <-> ( ph -> A. y ch ) ) )
11	10	pm5.74rd	|- ( A. x x = y -> ( ph -> ( A. x ps <-> A. y ch ) ) )
12	11	com12	|- ( ph -> ( A. x x = y -> ( A. x ps <-> A. y ch ) ) )