Theorem frege71 40635

Description: Lemma for frege72 40636. Proposition 71 of [Frege1879] p. 59. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 3-Jul-2020.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
frege71.x	⊢ 𝑋 ∈ 𝑉

Assertion

Ref	Expression
frege71	⊢ ((∀𝑧(𝑋𝑅𝑧 → 𝑧 ∈ 𝐴) → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)) → (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴))))

Distinct variable groups:   𝑧,𝐴   𝑧,𝑅   𝑧,𝑋

Allowed substitution hints:   𝑉(𝑧)   𝑌(𝑧)

Proof of Theorem frege71

Step	Hyp	Ref	Expression
1		frege71.x	. . 3 ⊢ 𝑋 ∈ 𝑉
2	1	frege70 40634	. 2 ⊢ (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → ∀𝑧(𝑋𝑅𝑧 → 𝑧 ∈ 𝐴)))
3		frege19 40525	. 2 ⊢ ((𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → ∀𝑧(𝑋𝑅𝑧 → 𝑧 ∈ 𝐴))) → ((∀𝑧(𝑋𝑅𝑧 → 𝑧 ∈ 𝐴) → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)) → (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)))))
4	2, 3	ax-mp 5	1 ⊢ ((∀𝑧(𝑋𝑅𝑧 → 𝑧 ∈ 𝐴) → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)) → (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴))))

Syntax hints:   → wi 4  ∀wal 1536   ∈ wcel 2111   class class class wbr 5030   hereditary whe 40473

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 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-frege1 40491  ax-frege2 40492  ax-frege8 40510  ax-frege52a 40558  ax-frege58b 40602

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-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-he 40474

This theorem is referenced by:  frege72  40636