Theorem frege55lem2b 43239

Description: Lemma for frege55b 43240. Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege55lem2b	⊢ (𝑥 = 𝑦 → [𝑦 / 𝑧]𝑧 = 𝑥)

Proof of Theorem frege55lem2b

Step	Hyp	Ref	Expression
1		frege54cor1b 43237	. 2 ⊢ [𝑥 / 𝑧]𝑧 = 𝑥
2		frege53b 43233	. 2 ⊢ ([𝑥 / 𝑧]𝑧 = 𝑥 → (𝑥 = 𝑦 → [𝑦 / 𝑧]𝑧 = 𝑥))
3	1, 2	ax-mp 5	1 ⊢ (𝑥 = 𝑦 → [𝑦 / 𝑧]𝑧 = 𝑥)

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 2164  ax-13 2366  ax-ext 2698  ax-frege8 43152  ax-frege52c 43231

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-sbc 3775

This theorem is referenced by:  frege55b  43240