Theorem frege55b 43859

Description: Lemma for frege57b 43861. Proposition 55 of [Frege1879] p. 50.

Note that eqtr2 2764 incorporates eqcom 2747 which is stronger than this proposition which is identical to equcomi 2016. 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 43858	. 2 ⊢ (𝑥 = 𝑦 → [𝑦 / 𝑧]𝑧 = 𝑥)
2		dfsb1 2489	. . 3 ⊢ ([𝑦 / 𝑧]𝑧 = 𝑥 ↔ ((𝑧 = 𝑦 → 𝑧 = 𝑥) ∧ ∃𝑧(𝑧 = 𝑦 ∧ 𝑧 = 𝑥)))
3		eqtr2 2764	. . . . 5 ⊢ ((𝑧 = 𝑦 ∧ 𝑧 = 𝑥) → 𝑦 = 𝑥)
4	3	exlimiv 1929	. . . 4 ⊢ (∃𝑧(𝑧 = 𝑦 ∧ 𝑧 = 𝑥) → 𝑦 = 𝑥)
5	4	adantl 481	. . 3 ⊢ (((𝑧 = 𝑦 → 𝑧 = 𝑥) ∧ ∃𝑧(𝑧 = 𝑦 ∧ 𝑧 = 𝑥)) → 𝑦 = 𝑥)
6	2, 5	sylbi 217	. 2 ⊢ ([𝑦 / 𝑧]𝑧 = 𝑥 → 𝑦 = 𝑥)
7	1, 6	syl 17	1 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1777  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-13 2380  ax-ext 2711  ax-frege8 43771  ax-frege52c 43850

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-sbc 3805

This theorem is referenced by:  frege56b  43860