Theorem frege56b 43602

Description: Lemma for frege57b 43603. Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege56b	⊢ ((𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)) → (𝑦 = 𝑥 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))

Proof of Theorem frege56b

Step	Hyp	Ref	Expression
1		frege55b 43601	. 2 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦)
2		frege9 43516	. 2 ⊢ ((𝑦 = 𝑥 → 𝑥 = 𝑦) → ((𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)) → (𝑦 = 𝑥 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))))
3	1, 2	ax-mp 5	1 ⊢ ((𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)) → (𝑦 = 𝑥 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))

Syntax hints:   → wi 4  [wsb 2060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-13 2366  ax-ext 2697  ax-frege1 43494  ax-frege2 43495  ax-frege8 43513  ax-frege52c 43592

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-sbc 3776

This theorem is referenced by:  frege57b  43603