Theorem frege52b 38183

Description: The case when the content of x is identical with the content of y and in which a proposition controlled by an element for which we substitute the content of x is affirmed and the same proposition, this time where we substitute the content of y, is denied does not take place. In [ x / z ] ph, x can also occur in other than the argument ( z) places. Hence x may still be contained in [ y / z ] ph. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege52b	|- ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) )

Proof of Theorem frege52b

Step	Hyp	Ref	Expression
1		ax-frege52c 38182	. 2 |- ( x = y -> ( [. x / z ]. ph -> [. y / z ]. ph ) )
2		sbsbc 3439	. 2 |- ( [ x / z ] ph <-> [. x / z ]. ph )
3		sbsbc 3439	. 2 |- ( [ y / z ] ph <-> [. y / z ]. ph )
4	1, 2, 3	3imtr4g 285	1 |- ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) )

Syntax hints:   -> wi 4   [wsb 1880   [.wsbc 3435

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-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:  frege53b  38184  frege57b  38193