Theorem frege55lem1b 38189

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

Assertion

Ref	Expression
frege55lem1b	|- ( ( ph -> [ x / y ] y = z ) -> ( ph -> x = z ) )

Distinct variable group:   y, z

Allowed substitution hints:   ph( x, y, z)

Proof of Theorem frege55lem1b

Step	Hyp	Ref	Expression
1		equsb3 2432	. . 3 |- ( [ x / y ] y = z <-> x = z )
2	1	biimpi 206	. 2 |- ( [ x / y ] y = z -> x = z )
3	2	imim2i 16	1 |- ( ( ph -> [ x / y ] y = z ) -> ( ph -> x = z ) )

Syntax hints:   -> wi 4   [wsb 1880

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-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by: (None)