Theorem frege112 44320

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 44313	. 2 ⊢ ((¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋) → 𝑋((t+‘𝑅) ∪ I )𝑍)
3		frege11 44159	. 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 1542   ∈ wcel 2114   ∪ cun 3901   class class class wbr 5100   I cid 5526  ‘cfv 6500  t+ctcl 14920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379  ax-frege1 44135  ax-frege2 44136  ax-frege8 44154  ax-frege52a 44202

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ifp 1064  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639

This theorem is referenced by:  frege113  44321  frege122  44330