Theorem frege55b 41878

Description: Lemma for frege57b 41880. Proposition 55 of [Frege1879] p. 50.

Note that eqtr2 2761 incorporates eqcom 2744 which is stronger than this proposition which is identical to equcomi 2020. Is it possible that Frege tricked himself into assuming what he was out to prove? (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege55b	⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)

Proof of Theorem frege55b

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frege55lem2b 41877	. 2 ⊢ (𝑥 = 𝑦 → [𝑦 / 𝑧]𝑧 = 𝑥)
2		dfsb1 2484	. . 3 ⊢ ([𝑦 / 𝑧]𝑧 = 𝑥 ↔ ((𝑧 = 𝑦 → 𝑧 = 𝑥) ∧ ∃𝑧(𝑧 = 𝑦 ∧ 𝑧 = 𝑥)))
3		eqtr2 2761	. . . . 5 ⊢ ((𝑧 = 𝑦 ∧ 𝑧 = 𝑥) → 𝑦 = 𝑥)
4	3	exlimiv 1933	. . . 4 ⊢ (∃𝑧(𝑧 = 𝑦 ∧ 𝑧 = 𝑥) → 𝑦 = 𝑥)
5	4	adantl 483	. . 3 ⊢ (((𝑧 = 𝑦 → 𝑧 = 𝑥) ∧ ∃𝑧(𝑧 = 𝑦 ∧ 𝑧 = 𝑥)) → 𝑦 = 𝑥)
6	2, 5	sylbi 216	. 2 ⊢ ([𝑦 / 𝑧]𝑧 = 𝑥 → 𝑦 = 𝑥)
7	1, 6	syl 17	1 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)

Syntax hints:   → wi 4   ∧ wa 397  ∃wex 1781  [wsb 2067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-13 2371  ax-ext 2708  ax-frege8 41790  ax-frege52c 41869

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-sbc 3731

This theorem is referenced by:  frege56b  41879