Theorem frege71 44290

Description: Lemma for frege72 44291. 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 44289	. 2 ⊢ (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → ∀𝑧(𝑋𝑅𝑧 → 𝑧 ∈ 𝐴)))
3		frege19 44180	. 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 1540   ∈ wcel 2114   class class class wbr 5100   hereditary whe 44128

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pr 5379  ax-frege1 44146  ax-frege2 44147  ax-frege8 44165  ax-frege52a 44213  ax-frege58b 44257

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ifp 1064  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5638  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-he 44129

This theorem is referenced by:  frege72  44291