Theorem frege56b 38192

Description: Lemma for frege57b 38193. Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege56b	|- ( ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) -> ( y = x -> ( [ x / z ] ph -> [ y / z ] ph ) ) )

Proof of Theorem frege56b

Step	Hyp	Ref	Expression
1		frege55b 38191	. 2 |- ( y = x -> x = y )
2		frege9 38106	. 2 |- ( ( y = x -> x = y ) -> ( ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) -> ( y = x -> ( [ x / z ] ph -> [ y / z ] ph ) ) ) )
3	1, 2	ax-mp 5	1 |- ( ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) -> ( y = x -> ( [ x / z ] ph -> [ y / z ] ph ) ) )

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-9 1999  ax-12 2047  ax-13 2246  ax-ext 2602  ax-frege1 38084  ax-frege2 38085  ax-frege8 38103  ax-frege52c 38182

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-sbc 3436

This theorem is referenced by:  frege57b  38193