Theorem frege107 43974

Description: Proposition 107 of [Frege1879] p. 74. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
frege107.v	⊢ 𝑉 ∈ 𝐴

Assertion

Ref	Expression
frege107	⊢ ((𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉 → 𝑍(t+‘𝑅)𝑉)) → (𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉 → 𝑍((t+‘𝑅) ∪ I )𝑉)))

Proof of Theorem frege107

Step	Hyp	Ref	Expression
1		frege107.v	. . 3 ⊢ 𝑉 ∈ 𝐴
2	1	frege106 43973	. 2 ⊢ (𝑍(t+‘𝑅)𝑉 → 𝑍((t+‘𝑅) ∪ I )𝑉)
3		frege7 43812	. 2 ⊢ ((𝑍(t+‘𝑅)𝑉 → 𝑍((t+‘𝑅) ∪ I )𝑉) → ((𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉 → 𝑍(t+‘𝑅)𝑉)) → (𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉 → 𝑍((t+‘𝑅) ∪ I )𝑉))))
4	2, 3	ax-mp 5	1 ⊢ ((𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉 → 𝑍(t+‘𝑅)𝑉)) → (𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉 → 𝑍((t+‘𝑅) ∪ I )𝑉)))

Syntax hints:   → wi 4   ∈ wcel 2107   ∪ cun 3962   class class class wbr 5149   I cid 5583  ‘cfv 6566  t+ctcl 15027

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 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5303  ax-nul 5313  ax-pr 5439  ax-frege1 43794  ax-frege2 43795  ax-frege8 43813  ax-frege28 43834  ax-frege31 43838  ax-frege52a 43861

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3an 1088  df-tru 1541  df-fal 1551  df-ex 1778  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3435  df-v 3481  df-dif 3967  df-un 3969  df-ss 3981  df-nul 4341  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5150  df-opab 5212  df-id 5584  df-xp 5696  df-rel 5697

This theorem is referenced by:  frege108  43975