Theorem frege104 42718

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 3494	. . 3 ⊢ 𝑍 ∈ V
3		eqeq1 2737	. . . 4 ⊢ (𝑧 = 𝑍 → (𝑧 = 𝑋 ↔ 𝑍 = 𝑋))
4		eqeq2 2745	. . . 4 ⊢ (𝑧 = 𝑍 → (𝑋 = 𝑧 ↔ 𝑋 = 𝑍))
5	3, 4	imbi12d 345	. . 3 ⊢ (𝑧 = 𝑍 → ((𝑧 = 𝑋 → 𝑋 = 𝑧) ↔ (𝑍 = 𝑋 → 𝑋 = 𝑍)))
6		frege55c 42669	. . 3 ⊢ (𝑧 = 𝑋 → 𝑋 = 𝑧)
7	2, 5, 6	vtocl 3550	. 2 ⊢ (𝑍 = 𝑋 → 𝑋 = 𝑍)
8	1	frege103 42717	. 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 1542   ∈ wcel 2107   ∪ cun 3947   class class class wbr 5149   I cid 5574  ‘cfv 6544  t+ctcl 14932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-frege1 42541  ax-frege2 42542  ax-frege8 42560  ax-frege52a 42608  ax-frege52c 42639

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ifp 1063  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684

This theorem is referenced by:  frege114  42728