Theorem elimhyps2

Description: Generalization of elimhyps that is not useful unless we can separately prove |- A e. _V . (Contributed by NM, 13-Jun-2019)

	Ref	Expression
Hypothesis	elimhyps2.1	|- [. B / x ]. ph
Assertion	elimhyps2	|- [. if ( [. A / x ]. ph , A , B ) / x ]. ph

Proof

Step	Hyp	Ref	Expression
1		elimhyps2.1	|- [. B / x ]. ph
2		dfsbcq	|- ( A = if ( [. A / x ]. ph , A , B ) -> ( [. A / x ]. ph <-> [. if ( [. A / x ]. ph , A , B ) / x ]. ph ) )
3		dfsbcq	|- ( B = if ( [. A / x ]. ph , A , B ) -> ( [. B / x ]. ph <-> [. if ( [. A / x ]. ph , A , B ) / x ]. ph ) )
4	2 3 1	elimhyp	|- [. if ( [. A / x ]. ph , A , B ) / x ]. ph