Theorem frege113 41894

Description: Proposition 113 of [Frege1879] p. 76. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
frege112.z	⊢ 𝑍 ∈ 𝑉

Assertion

Ref	Expression
frege113	⊢ ((𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑍 = 𝑋)) → (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑋((t+‘𝑅) ∪ I )𝑍)))

Proof of Theorem frege113

Step	Hyp	Ref	Expression
1		frege112.z	. . 3 ⊢ 𝑍 ∈ 𝑉
2	1	frege112 41893	. 2 ⊢ (𝑍 = 𝑋 → 𝑋((t+‘𝑅) ∪ I )𝑍)
3		frege7 41726	. 2 ⊢ ((𝑍 = 𝑋 → 𝑋((t+‘𝑅) ∪ I )𝑍) → ((𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑍 = 𝑋)) → (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑋((t+‘𝑅) ∪ I )𝑍))))
4	2, 3	ax-mp 5	1 ⊢ ((𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑍 = 𝑋)) → (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑋((t+‘𝑅) ∪ I )𝑍)))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2105   ∪ cun 3895   class class class wbr 5089   I cid 5511  ‘cfv 6473  t+ctcl 14787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-sep 5240  ax-nul 5247  ax-pr 5369  ax-frege1 41708  ax-frege2 41709  ax-frege8 41727  ax-frege52a 41775

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ifp 1061  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-sn 4573  df-pr 4575  df-op 4579  df-br 5090  df-opab 5152  df-id 5512  df-xp 5620  df-rel 5621

This theorem is referenced by:  frege114  41895