Theorem frege53b 40591

Description: Lemma for frege102 (via frege92 40656). Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege53b	⊢ ([𝑦 / 𝑥]𝜑 → (𝑦 = 𝑧 → [𝑧 / 𝑥]𝜑))

Proof of Theorem frege53b

Step	Hyp	Ref	Expression
1		frege52b 40590	. 2 ⊢ (𝑦 = 𝑧 → ([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑))
2		ax-frege8 40510	. 2 ⊢ ((𝑦 = 𝑧 → ([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑)) → ([𝑦 / 𝑥]𝜑 → (𝑦 = 𝑧 → [𝑧 / 𝑥]𝜑)))
3	1, 2	ax-mp 5	1 ⊢ ([𝑦 / 𝑥]𝜑 → (𝑦 = 𝑧 → [𝑧 / 𝑥]𝜑))

Syntax hints:   → wi 4  [wsb 2069

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770  ax-frege8 40510  ax-frege52c 40589

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-clab 2777  df-cleq 2791  df-clel 2870  df-sbc 3721

This theorem is referenced by:  frege55lem2b  40597