Theorem frege104 44275

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 3464	. . 3 ⊢ 𝑍 ∈ V
3		eqeq1 2741	. . . 4 ⊢ (𝑧 = 𝑍 → (𝑧 = 𝑋 ↔ 𝑍 = 𝑋))
4		eqeq2 2749	. . . 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 44226	. . 3 ⊢ (𝑧 = 𝑋 → 𝑋 = 𝑧)
7	2, 5, 6	vtocl 3516	. 2 ⊢ (𝑍 = 𝑋 → 𝑋 = 𝑍)
8	1	frege103 44274	. 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 2114   ∪ cun 3900   class class class wbr 5099   I cid 5519  ‘cfv 6493  t+ctcl 14912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-frege1 44098  ax-frege2 44099  ax-frege8 44117  ax-frege52a 44165  ax-frege52c 44196

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ifp 1064  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-sbc 3742  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-id 5520  df-xp 5631  df-rel 5632

This theorem is referenced by:  frege114  44285