Theorem frege56b 43207

Description: Lemma for frege57b 43208. 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 43206	. 2 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦)
2		frege9 43121	. 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 2059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-13 2365  ax-ext 2697  ax-frege1 43099  ax-frege2 43100  ax-frege8 43118  ax-frege52c 43197

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-sbc 3773

This theorem is referenced by:  frege57b  43208