Theorem frege100 43925

Description: One direction of dffrege99 43924. Proposition 100 of [Frege1879] p. 72. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
frege99.z	⊢ 𝑍 ∈ 𝑈

Assertion

Ref	Expression
frege100	⊢ (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋))

Proof of Theorem frege100

Step	Hyp	Ref	Expression
1		frege99.z	. . 3 ⊢ 𝑍 ∈ 𝑈
2	1	dffrege99 43924	. 2 ⊢ ((¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋) ↔ 𝑋((t+‘𝑅) ∪ I )𝑍)
3		frege57aid 43834	. 2 ⊢ (((¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋) ↔ 𝑋((t+‘𝑅) ∪ I )𝑍) → (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋)))
4	2, 3	ax-mp 5	1 ⊢ (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2108   ∪ cun 3974   class class class wbr 5166   I cid 5592  ‘cfv 6573  t+ctcl 15034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-frege1 43752  ax-frege2 43753  ax-frege8 43771  ax-frege52a 43819

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ifp 1064  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707

This theorem is referenced by:  frege101  43926  frege103  43928