Theorem frege100 43952

Description: One direction of dffrege99 43951. 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 43951	. 2 ⊢ ((¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋) ↔ 𝑋((t+‘𝑅) ∪ I )𝑍)
3		frege57aid 43861	. 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 1540   ∈ wcel 2109   ∪ cun 3912   class class class wbr 5107   I cid 5532  ‘cfv 6511  t+ctcl 14951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-frege1 43779  ax-frege2 43780  ax-frege8 43798  ax-frege52a 43846

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645

This theorem is referenced by:  frege101  43953  frege103  43955