Theorem frege52b 42630

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]𝜑, 𝑥 can also occur in other than the argument (𝑧) places. Hence 𝑥 may still be contained in [𝑦 / 𝑧]𝜑. 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]𝜑 → [𝑦 / 𝑧]𝜑))

Proof of Theorem frege52b

Step	Hyp	Ref	Expression
1		ax-frege52c 42629	. 2 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
2		sbsbc 3781	. 2 ⊢ ([𝑥 / 𝑧]𝜑 ↔ [𝑥 / 𝑧]𝜑)
3		sbsbc 3781	. 2 ⊢ ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑)
4	1, 2, 3	3imtr4g 295	1 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))

Syntax hints:   → wi 4  [wsb 2067  [wsbc 3777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-frege52c 42629

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-clab 2710  df-cleq 2724  df-clel 2810  df-sbc 3778

This theorem is referenced by:  frege53b  42631  frege57b  42640