Theorem frege104 43870

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 3506	. . 3 ⊢ 𝑍 ∈ V
3		eqeq1 2738	. . . 4 ⊢ (𝑧 = 𝑍 → (𝑧 = 𝑋 ↔ 𝑍 = 𝑋))
4		eqeq2 2746	. . . 4 ⊢ (𝑧 = 𝑍 → (𝑋 = 𝑧 ↔ 𝑋 = 𝑍))
5	3, 4	imbi12d 344	. . 3 ⊢ (𝑧 = 𝑍 → ((𝑧 = 𝑋 → 𝑋 = 𝑧) ↔ (𝑍 = 𝑋 → 𝑋 = 𝑍)))
6		frege55c 43821	. . 3 ⊢ (𝑧 = 𝑋 → 𝑋 = 𝑧)
7	2, 5, 6	vtocl 3565	. 2 ⊢ (𝑍 = 𝑋 → 𝑋 = 𝑍)
8	1	frege103 43869	. 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))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1537   ∈ wcel 2103   ∪ cun 3968   class class class wbr 5169   I cid 5596  ‘cfv 6572  t+ctcl 15030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450  ax-frege1 43693  ax-frege2 43694  ax-frege8 43712  ax-frege52a 43760  ax-frege52c 43791

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ifp 1064  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-sbc 3799  df-dif 3973  df-un 3975  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5170  df-opab 5232  df-id 5597  df-xp 5705  df-rel 5706

This theorem is referenced by:  frege114  43880