Theorem frege56b 39031

Description: Lemma for frege57b 39032. 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 39030	. 2 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦)
2		frege9 38945	. 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 2067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-12 2220  ax-13 2389  ax-ext 2803  ax-frege1 38923  ax-frege2 38924  ax-frege8 38942  ax-frege52c 39021

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1879  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-sbc 3663

This theorem is referenced by:  frege57b  39032