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Theorem frege123 38803
Description: Lemma for frege124 38804. Proposition 123 of [Frege1879] p. 79. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
frege123.x 𝑋𝑈
frege123.y 𝑌𝑉
Assertion
Ref Expression
frege123 ((∀𝑎(𝑌𝑅𝑎𝑋((t+‘𝑅) ∪ I )𝑎) → (𝑌(t+‘𝑅)𝑀𝑋((t+‘𝑅) ∪ I )𝑀)) → (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌(t+‘𝑅)𝑀𝑋((t+‘𝑅) ∪ I )𝑀))))
Distinct variable groups:   𝑅,𝑎   𝑋,𝑎   𝑌,𝑎
Allowed substitution hints:   𝑈(𝑎)   𝑀(𝑎)   𝑉(𝑎)

Proof of Theorem frege123
StepHypRef Expression
1 frege123.x . . . 4 𝑋𝑈
2 frege123.y . . . 4 𝑌𝑉
3 vex 3354 . . . 4 𝑎 ∈ V
41, 2, 3frege122 38802 . . 3 (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝑎𝑋((t+‘𝑅) ∪ I )𝑎)))
54alrimdv 2009 . 2 (Fun 𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑋((t+‘𝑅) ∪ I )𝑎)))
6 frege19 38641 . 2 ((Fun 𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑋((t+‘𝑅) ∪ I )𝑎))) → ((∀𝑎(𝑌𝑅𝑎𝑋((t+‘𝑅) ∪ I )𝑎) → (𝑌(t+‘𝑅)𝑀𝑋((t+‘𝑅) ∪ I )𝑀)) → (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌(t+‘𝑅)𝑀𝑋((t+‘𝑅) ∪ I )𝑀)))))
75, 6ax-mp 5 1 ((∀𝑎(𝑌𝑅𝑎𝑋((t+‘𝑅) ∪ I )𝑎) → (𝑌(t+‘𝑅)𝑀𝑋((t+‘𝑅) ∪ I )𝑀)) → (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌(t+‘𝑅)𝑀𝑋((t+‘𝑅) ∪ I )𝑀))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1629  wcel 2145  Vcvv 3351  cun 3721   class class class wbr 4786   I cid 5156  ccnv 5248  Fun wfun 6023  cfv 6029  t+ctcl 13930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-frege1 38607  ax-frege2 38608  ax-frege8 38626  ax-frege52a 38674  ax-frege58b 38718
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-ifp 1050  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-fun 6031
This theorem is referenced by:  frege124  38804
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