Theorem dedths2 34251

Description: Generalization of dedths 34248 that is not useful unless we can separately prove |- A e. _V. (Contributed by NM, 13-Jun-2019.)

Hypothesis

Ref	Expression
dedths2.1	|- [. if ( [. A / x ]. ph , A , B ) / x ]. ps

Assertion

Ref	Expression
dedths2	|- ( [. A / x ]. ph -> [. A / x ]. ps )

Proof of Theorem dedths2

Step	Hyp	Ref	Expression
1		dfsbcq 3437	. 2 |- ( A = if ( [. A / x ]. ph , A , B ) -> ( [. A / x ]. ps <-> [. if ( [. A / x ]. ph , A , B ) / x ]. ps ) )
2		dedths2.1	. 2 |- [. if ( [. A / x ]. ph , A , B ) / x ]. ps
3	1, 2	dedth 4139	1 |- ( [. A / x ]. ph -> [. A / x ]. ps )

Syntax hints:   -> wi 4   [.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: (None)