Theorem frege104 45009

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 3472	. . 3 ⊢ 𝑍 ∈ V
3		eqeq1 2764	. . . 4 ⊢ (𝑧 = 𝑍 → (𝑧 = 𝑋 ↔ 𝑍 = 𝑋))
4		eqeq2 2772	. . . 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 44960	. . 3 ⊢ (𝑧 = 𝑋 → 𝑋 = 𝑧)
7	2, 5, 6	vtocl 3520	. 2 ⊢ (𝑍 = 𝑋 → 𝑋 = 𝑍)
8	1	frege103 45008	. 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 2144   ∪ cun 4149   class class class wbr 5102   I cid 5544  ‘cfv 6538  t+ctcl 15138

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390  ax-frege1 44832  ax-frege2 44833  ax-frege8 44851  ax-frege52a 44899  ax-frege52c 44930

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657

This theorem is used by:  frege114  45019