Theorem dedths 34248

Description: A version of weak deduction theorem dedth 4139 using explicit substitution. (Contributed by NM, 15-Jun-2019.)

Hypothesis

Ref	Expression
dedths.1	|- [. if ( ph , x , B ) / x ]. ps

Assertion

Ref	Expression
dedths	|- ( ph -> ps )

Proof of Theorem dedths

Step	Hyp	Ref	Expression
1		dfsbcq 3437	. . 3 |- ( x = if ( [. x / x ]. ph , x , B ) -> ( [. x / x ]. ps <-> [. if ( [. x / x ]. ph , x , B ) / x ]. ps ) )
2		dedths.1	. . . 4 |- [. if ( ph , x , B ) / x ]. ps
3		sbcid 3452	. . . . 5 |- ( [. x / x ]. ph <-> ph )
4		ifbi 4107	. . . . 5 |- ( ( [. x / x ]. ph <-> ph ) -> if ( [. x / x ]. ph , x , B ) = if ( ph , x , B ) )
5		dfsbcq 3437	. . . . 5 |- ( if ( [. x / x ]. ph , x , B ) = if ( ph , x , B ) -> ( [. if ( [. x / x ]. ph , x , B ) / x ]. ps <-> [. if ( ph , x , B ) / x ]. ps ) )
6	3, 4, 5	mp2b 10	. . . 4 |- ( [. if ( [. x / x ]. ph , x , B ) / x ]. ps <-> [. if ( ph , x , B ) / x ]. ps )
7	2, 6	mpbir 221	. . 3 |- [. if ( [. x / x ]. ph , x , B ) / x ]. ps
8	1, 7	dedth 4139	. 2 |- ( [. x / x ]. ph -> [. x / x ]. ps )
9		sbcid 3452	. 2 |- ( [. x / x ]. ps <-> ps )
10	8, 3, 9	3imtr3i 280	1 |- ( ph -> ps )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   [.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