Theorem frege107 43941

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 43940	. 2 ⊢ (𝑍(t+‘𝑅)𝑉 → 𝑍((t+‘𝑅) ∪ I )𝑉)
3		frege7 43779	. 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 2108   ∪ cun 3924   class class class wbr 5119   I cid 5547  ‘cfv 6530  t+ctcl 15002

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-frege1 43761  ax-frege2 43762  ax-frege8 43780  ax-frege28 43801  ax-frege31 43805  ax-frege52a 43828

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 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661

This theorem is referenced by:  frege108  43942