Theorem frege55b 40249

Description: Lemma for frege57b 40251. Proposition 55 of [Frege1879] p. 50.

Note that eqtr2 2845 incorporates eqcom 2831 which is stronger than this proposition which is identical to equcomi 2023. 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 40248	. 2 ⊢ (𝑥 = 𝑦 → [𝑦 / 𝑧]𝑧 = 𝑥)
2		dfsb1 2509	. . 3 ⊢ ([𝑦 / 𝑧]𝑧 = 𝑥 ↔ ((𝑧 = 𝑦 → 𝑧 = 𝑥) ∧ ∃𝑧(𝑧 = 𝑦 ∧ 𝑧 = 𝑥)))
3		eqtr2 2845	. . . . 5 ⊢ ((𝑧 = 𝑦 ∧ 𝑧 = 𝑥) → 𝑦 = 𝑥)
4	3	exlimiv 1930	. . . 4 ⊢ (∃𝑧(𝑧 = 𝑦 ∧ 𝑧 = 𝑥) → 𝑦 = 𝑥)
5	4	adantl 484	. . 3 ⊢ (((𝑧 = 𝑦 → 𝑧 = 𝑥) ∧ ∃𝑧(𝑧 = 𝑦 ∧ 𝑧 = 𝑥)) → 𝑦 = 𝑥)
6	2, 5	sylbi 219	. 2 ⊢ ([𝑦 / 𝑧]𝑧 = 𝑥 → 𝑦 = 𝑥)
7	1, 6	syl 17	1 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)

Syntax hints:   → wi 4   ∧ wa 398  ∃wex 1779  [wsb 2068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-12 2176  ax-13 2389  ax-ext 2796  ax-frege8 40161  ax-frege52c 40240

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-sbc 3776

This theorem is referenced by:  frege56b  40250