Theorem frege104 44721

Description: Proposition 104 of [Frege1879] p. 73.

Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collection From Frege to Goedel, this proof has the minor clause and result swapped. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
frege103.z	⊢ 𝑍 ∈ 𝑉

Assertion

Ref	Expression
frege104	⊢ (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍 → 𝑋 = 𝑍))

Proof of Theorem frege104

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frege103.z	. . . 4 ⊢ 𝑍 ∈ 𝑉
2	1	elexi 3476	. . 3 ⊢ 𝑍 ∈ V
3		eqeq1 2766	. . . 4 ⊢ (𝑧 = 𝑍 → (𝑧 = 𝑋 ↔ 𝑍 = 𝑋))
4		eqeq2 2774	. . . 4 ⊢ (𝑧 = 𝑍 → (𝑋 = 𝑧 ↔ 𝑋 = 𝑍))
5	3, 4	imbi12d 347	. . 3 ⊢ (𝑧 = 𝑍 → ((𝑧 = 𝑋 → 𝑋 = 𝑧) ↔ (𝑍 = 𝑋 → 𝑋 = 𝑍)))
6		frege55c 44672	. . 3 ⊢ (𝑧 = 𝑋 → 𝑋 = 𝑧)
7	2, 5, 6	vtocl 3524	. 2 ⊢ (𝑍 = 𝑋 → 𝑋 = 𝑍)
8	1	frege103 44720	. 2 ⊢ ((𝑍 = 𝑋 → 𝑋 = 𝑍) → (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍 → 𝑋 = 𝑍)))
9	7, 8	ax-mp 5	1 ⊢ (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍 → 𝑋 = 𝑍))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1569   ∈ wcel 2142   ∪ cun 3902   class class class wbr 5108   I cid 5554  ‘cfv 6536  t+ctcl 15029

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-pr 5403  ax-frege1 44544  ax-frege2 44545  ax-frege8 44563  ax-frege52a 44611  ax-frege52c 44642

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ifp 1078  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-id 5555  df-xp 5666  df-rel 5667

This theorem is used by:  frege114  44731