Theorem frege53b 39025

Description: Lemma for frege102 (via frege92 39090). Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem frege53b

Step	Hyp	Ref	Expression
1		frege52b 39024	. 2 ⊢ (𝑥 = 𝑧 → ([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜑))
2		ax-frege8 38944	. 2 ⊢ ((𝑥 = 𝑧 → ([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜑)) → ([𝑥 / 𝑦]𝜑 → (𝑥 = 𝑧 → [𝑧 / 𝑦]𝜑)))
3	1, 2	ax-mp 5	1 ⊢ ([𝑥 / 𝑦]𝜑 → (𝑥 = 𝑧 → [𝑧 / 𝑦]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 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-ext 2804  ax-frege8 38944  ax-frege52c 39023

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1881  df-clab 2813  df-cleq 2819  df-clel 2822  df-sbc 3664

This theorem is referenced by:  frege55lem2b  39031