Theorem frege56b 37212

Description: Lemma for frege57b 37213. 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 37211	. 2 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦)
2		frege9 37126	. 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 1867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-13 2234  ax-ext 2590  ax-frege1 37104  ax-frege2 37105  ax-frege8 37123  ax-frege52c 37202

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-sbc 3403

This theorem is referenced by:  frege57b  37213