Theorem frege55lem1a 38160

Description: Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege55lem1a	|- ( ( ta -> if- ( ps , ph , -. ph ) ) -> ( ta -> ( ps <-> ph ) ) )

Proof of Theorem frege55lem1a

Step	Hyp	Ref	Expression
1		frege54cor0a 38157	. . 3 |- ( ( ps <-> ph ) <-> if- ( ps , ph , -. ph ) )
2	1	biimpri 218	. 2 |- (if- ( ps , ph , -. ph ) -> ( ps <-> ph ) )
3	2	imim2i 16	1 |- ( ( ta -> if- ( ps , ph , -. ph ) ) -> ( ta -> ( ps <-> ph ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196  if-wif 1012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege28 38124

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013

This theorem is referenced by:  frege55cor1a  38163