Theorem frege53b 38184

Description: Lemma for frege102 (via frege92 38249). Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege53b	|- ( [ x / y ] ph -> ( x = z -> [ z / y ] ph ) )

Proof of Theorem frege53b

Step	Hyp	Ref	Expression
1		frege52b 38183	. 2 |- ( x = z -> ( [ x / y ] ph -> [ z / y ] ph ) )
2		ax-frege8 38103	. 2 |- ( ( x = z -> ( [ x / y ] ph -> [ z / y ] ph ) ) -> ( [ x / y ] ph -> ( x = z -> [ z / y ] ph ) ) )
3	1, 2	ax-mp 5	1 |- ( [ x / y ] ph -> ( x = z -> [ z / y ] 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-ext 2602  ax-frege8 38103  ax-frege52c 38182

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

This theorem is referenced by:  frege55lem2b  38190