Theorem frege74 44881

Description: If 𝑋 has a property 𝐴 that is hereditary in the 𝑅-sequence, then every result of a application of the procedure 𝑅 to 𝑋 has the property 𝐴. Proposition 74 of [Frege1879] p. 60. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
frege74.x	⊢ 𝑋 ∈ 𝑈
frege74.y	⊢ 𝑌 ∈ 𝑉

Assertion

Ref	Expression
frege74	⊢ (𝑋 ∈ 𝐴 → (𝑅 hereditary 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)))

Proof of Theorem frege74

Step	Hyp	Ref	Expression
1		frege74.x	. . 3 ⊢ 𝑋 ∈ 𝑈
2		frege74.y	. . 3 ⊢ 𝑌 ∈ 𝑉
3	1, 2	frege72 44879	. 2 ⊢ (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)))
4		ax-frege8 44753	. 2 ⊢ ((𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴))) → (𝑋 ∈ 𝐴 → (𝑅 hereditary 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴))))
5	3, 4	ax-mp 5	1 ⊢ (𝑋 ∈ 𝐴 → (𝑅 hereditary 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)))

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   class class class wbr 5102   hereditary whe 44716

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-pr 5390  ax-frege1 44734  ax-frege2 44735  ax-frege8 44753  ax-frege52a 44801  ax-frege58b 44845

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5653  df-cnv 5655  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-he 44717

This theorem is used by:  frege81  44888