Theorem frege107 44419

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 44418	. 2 ⊢ (𝑍(t+‘𝑅)𝑉 → 𝑍((t+‘𝑅) ∪ I )𝑉)
3		frege7 44257	. 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 2114   ∪ cun 3888   class class class wbr 5086   I cid 5520  ‘cfv 6494  t+ctcl 14942

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 5232  ax-pr 5372  ax-frege1 44239  ax-frege2 44240  ax-frege8 44258  ax-frege28 44279  ax-frege31 44283  ax-frege52a 44306

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 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-id 5521  df-xp 5632  df-rel 5633

This theorem is referenced by:  frege108  44420