Theorem frege55c 45108

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 3458	. . . 4 ⊢ 𝑥 ∈ V
2	1	frege54cor1c 45105	. . 3 ⊢ [𝑥 / 𝑦]𝑦 = 𝑥
3		frege53c 45104	. . 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 3743	. . . 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 278	. . 3 ⊢ ([𝐴 / 𝑦]𝑦 = 𝑥 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = 𝑥))
8		eqtr2 2784	. . . 4 ⊢ ((𝑦 = 𝐴 ∧ 𝑦 = 𝑥) → 𝐴 = 𝑥)
9	8	exlimiv 1965	. . 3 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 = 𝑥) → 𝐴 = 𝑥)
10	7, 9	sylbi 220	. 2 ⊢ ([𝐴 / 𝑦]𝑦 = 𝑥 → 𝐴 = 𝑥)
11	4, 10	syl 18	1 ⊢ (𝑥 = 𝐴 → 𝐴 = 𝑥)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741  Vcvv 3454  [wsbc 3742

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-12 2216  ax-ext 2735  ax-frege8 44999  ax-frege52c 45078

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-sbc 3743  df-sn 4589

This theorem is used by:  frege104  45157