Theorem frege55c 44499

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

Assertion

Ref	Expression
frege55c	⊢ (𝑥 = 𝐴 → 𝐴 = 𝑥)

Proof of Theorem frege55c

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3460	. . . 4 ⊢ 𝑥 ∈ V
2	1	frege54cor1c 44496	. . 3 ⊢ [𝑥 / 𝑦]𝑦 = 𝑥
3		frege53c 44495	. . 3 ⊢ ([𝑥 / 𝑦]𝑦 = 𝑥 → (𝑥 = 𝐴 → [𝐴 / 𝑦]𝑦 = 𝑥))
4	2, 3	ax-mp 5	. 2 ⊢ (𝑥 = 𝐴 → [𝐴 / 𝑦]𝑦 = 𝑥)
5		df-sbc 3747	. . . 4 ⊢ ([𝐴 / 𝑦]𝑦 = 𝑥 ↔ 𝐴 ∈ {𝑦 ∣ 𝑦 = 𝑥})
6		clelab 2908	. . . 4 ⊢ (𝐴 ∈ {𝑦 ∣ 𝑦 = 𝑥} ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = 𝑥))
7	5, 6	bitri 277	. . 3 ⊢ ([𝐴 / 𝑦]𝑦 = 𝑥 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = 𝑥))
8		eqtr2 2785	. . . 4 ⊢ ((𝑦 = 𝐴 ∧ 𝑦 = 𝑥) → 𝐴 = 𝑥)
9	8	exlimiv 1952	. . 3 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = 𝑥) → 𝐴 = 𝑥)
10	7, 9	sylbi 219	. 2 ⊢ ([𝐴 / 𝑦]𝑦 = 𝑥 → 𝐴 = 𝑥)
11	4, 10	syl 17	1 ⊢ (𝑥 = 𝐴 → 𝐴 = 𝑥)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562  ∃wex 1801   ∈ wcel 2144  {cab 2742  Vcvv 3456  [wsbc 3746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-12 2214  ax-ext 2736  ax-frege8 44390  ax-frege52c 44469

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1565  df-ex 1802  df-nf 1806  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-v 3458  df-sbc 3747  df-sn 4585

This theorem is referenced by:  frege104  44548