Theorem elimhyps 34247

Description: A version of elimhyp 4146 using explicit substitution. (Contributed by NM, 15-Jun-2019.)

Hypothesis

Ref	Expression
elimhyps.1	|- [. B / x ]. ph

Assertion

Ref	Expression
elimhyps	|- [. if ( ph , x , B ) / x ]. ph

Proof of Theorem elimhyps

Step	Hyp	Ref	Expression
1		sbceq1a 3446	. 2 |- ( x = if ( ph , x , B ) -> ( ph <-> [. if ( ph , x , B ) / x ]. ph ) )
2		dfsbcq 3437	. 2 |- ( B = if ( ph , x , B ) -> ( [. B / x ]. ph <-> [. if ( ph , x , B ) / x ]. ph ) )
3		elimhyps.1	. 2 |- [. B / x ]. ph
4	1, 2, 3	elimhyp 4146	1 |- [. if ( ph , x , B ) / x ]. ph

Syntax hints:   [.wsbc 3435   ifcif 4086

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-sbc 3436  df-if 4087

This theorem is referenced by:  renegclALT  34249