Theorem frege55b 43886

Description: Lemma for frege57b 43888. Proposition 55 of [Frege1879] p. 50.

Note that eqtr2 2750 incorporates eqcom 2736 which is stronger than this proposition which is identical to equcomi 2017. 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 43885	. 2 ⊢ (𝑥 = 𝑦 → [𝑦 / 𝑧]𝑧 = 𝑥)
2		dfsb1 2479	. . 3 ⊢ ([𝑦 / 𝑧]𝑧 = 𝑥 ↔ ((𝑧 = 𝑦 → 𝑧 = 𝑥) ∧ ∃𝑧(𝑧 = 𝑦 ∧ 𝑧 = 𝑥)))
3		eqtr2 2750	. . . . 5 ⊢ ((𝑧 = 𝑦 ∧ 𝑧 = 𝑥) → 𝑦 = 𝑥)
4	3	exlimiv 1930	. . . 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 1779  [wsb 2065

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-13 2370  ax-ext 2701  ax-frege8 43798  ax-frege52c 43877

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-sbc 3754

This theorem is referenced by:  frege56b  43887