Theorem frege112 44671

Description: Identity implies belonging to the 𝑅-sequence beginning with self. Proposition 112 of [Frege1879] p. 76. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
frege112.z	⊢ 𝑍 ∈ 𝑉

Assertion

Ref	Expression
frege112	⊢ (𝑍 = 𝑋 → 𝑋((t+‘𝑅) ∪ I )𝑍)

Proof of Theorem frege112

Step	Hyp	Ref	Expression
1		frege112.z	. . 3 ⊢ 𝑍 ∈ 𝑉
2	1	frege105 44664	. 2 ⊢ ((¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋) → 𝑋((t+‘𝑅) ∪ I )𝑍)
3		frege11 44510	. 2 ⊢ (((¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋) → 𝑋((t+‘𝑅) ∪ I )𝑍) → (𝑍 = 𝑋 → 𝑋((t+‘𝑅) ∪ I )𝑍))
4	2, 3	ax-mp 5	1 ⊢ (𝑍 = 𝑋 → 𝑋((t+‘𝑅) ∪ I )𝑍)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1568   ∈ wcel 2141   ∪ cun 3902   class class class wbr 5108   I cid 5555  ‘cfv 6536  t+ctcl 15022

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-pr 5404  ax-frege1 44486  ax-frege2 44487  ax-frege8 44505  ax-frege52a 44553

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ifp 1077  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-id 5556  df-xp 5667  df-rel 5668

This theorem is referenced by:  frege113  44672  frege122  44681