Theorem frege74 41407

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 41405	. 2 ⊢ (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)))
4		ax-frege8 41279	. 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)))

Syntax hints:   → wi 4   ∈ wcel 2112   class class class wbr 5070   hereditary whe 41242

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-sep 5216  ax-nul 5223  ax-pr 5346  ax-frege1 41260  ax-frege2 41261  ax-frege8 41279  ax-frege52a 41327  ax-frege58b 41371

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-ifp 1064  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3425  df-sbc 3713  df-csb 3830  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4255  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5585  df-cnv 5587  df-dm 5589  df-rn 5590  df-res 5591  df-ima 5592  df-he 41243

This theorem is referenced by:  frege81  41414