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Theorem frege72 41044
 Description: If property 𝐴 is hereditary in the 𝑅-sequence, if 𝑥 has property 𝐴, and if 𝑦 is a result of an application of the procedure 𝑅 to 𝑥, then 𝑦 has property 𝐴. Proposition 72 of [Frege1879] p. 59. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
frege72.x 𝑋𝑈
frege72.y 𝑌𝑉
Assertion
Ref Expression
frege72 (𝑅 hereditary 𝐴 → (𝑋𝐴 → (𝑋𝑅𝑌𝑌𝐴)))

Proof of Theorem frege72
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 frege72.y . . . 4 𝑌𝑉
21frege58c 41030 . . 3 (∀𝑧(𝑋𝑅𝑧𝑧𝐴) → [𝑌 / 𝑧](𝑋𝑅𝑧𝑧𝐴))
3 sbcim1 3751 . . . 4 ([𝑌 / 𝑧](𝑋𝑅𝑧𝑧𝐴) → ([𝑌 / 𝑧]𝑋𝑅𝑧[𝑌 / 𝑧]𝑧𝐴))
4 sbcbr2g 5094 . . . . . 6 (𝑌𝑉 → ([𝑌 / 𝑧]𝑋𝑅𝑧𝑋𝑅𝑌 / 𝑧𝑧))
5 csbvarg 4331 . . . . . . 7 (𝑌𝑉𝑌 / 𝑧𝑧 = 𝑌)
65breq2d 5048 . . . . . 6 (𝑌𝑉 → (𝑋𝑅𝑌 / 𝑧𝑧𝑋𝑅𝑌))
74, 6bitrd 282 . . . . 5 (𝑌𝑉 → ([𝑌 / 𝑧]𝑋𝑅𝑧𝑋𝑅𝑌))
81, 7ax-mp 5 . . . 4 ([𝑌 / 𝑧]𝑋𝑅𝑧𝑋𝑅𝑌)
9 sbcel1v 3765 . . . 4 ([𝑌 / 𝑧]𝑧𝐴𝑌𝐴)
103, 8, 93imtr3g 298 . . 3 ([𝑌 / 𝑧](𝑋𝑅𝑧𝑧𝐴) → (𝑋𝑅𝑌𝑌𝐴))
112, 10syl 17 . 2 (∀𝑧(𝑋𝑅𝑧𝑧𝐴) → (𝑋𝑅𝑌𝑌𝐴))
12 frege72.x . . 3 𝑋𝑈
1312frege71 41043 . 2 ((∀𝑧(𝑋𝑅𝑧𝑧𝐴) → (𝑋𝑅𝑌𝑌𝐴)) → (𝑅 hereditary 𝐴 → (𝑋𝐴 → (𝑋𝑅𝑌𝑌𝐴))))
1411, 13ax-mp 5 1 (𝑅 hereditary 𝐴 → (𝑋𝐴 → (𝑋𝑅𝑌𝑌𝐴)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   ∈ wcel 2111  [wsbc 3698  ⦋csb 3807   class class class wbr 5036   hereditary whe 40881 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302  ax-frege1 40899  ax-frege2 40900  ax-frege8 40918  ax-frege52a 40966  ax-frege58b 41010 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5037  df-opab 5099  df-xp 5534  df-cnv 5536  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-he 40882 This theorem is referenced by:  frege73  41045  frege74  41046
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