Theorem frege113 44217

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 44216	. 2 ⊢ (𝑍 = 𝑋 → 𝑋((t+‘𝑅) ∪ I )𝑍)
3		frege7 44049	. 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 1541   ∈ wcel 2113   ∪ cun 3899   class class class wbr 5098   I cid 5518  ‘cfv 6492  t+ctcl 14908

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-frege1 44031  ax-frege2 44032  ax-frege8 44050  ax-frege52a 44098

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631

This theorem is referenced by:  frege114  44218