Theorem frege73 44740

Description: Lemma for frege87 44754. Proposition 73 of [Frege1879] p. 59. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
frege73.x	⊢ 𝑋 ∈ 𝑈
frege73.y	⊢ 𝑌 ∈ 𝑉

Assertion

Ref	Expression
frege73	⊢ ((𝑅 hereditary 𝐴 → 𝑋 ∈ 𝐴) → (𝑅 hereditary 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)))

Proof of Theorem frege73

Step	Hyp	Ref	Expression
1		frege73.x	. . 3 ⊢ 𝑋 ∈ 𝑈
2		frege73.y	. . 3 ⊢ 𝑌 ∈ 𝑉
3	1, 2	frege72 44739	. 2 ⊢ (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)))
4		ax-frege2 44595	. 2 ⊢ ((𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴))) → ((𝑅 hereditary 𝐴 → 𝑋 ∈ 𝐴) → (𝑅 hereditary 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴))))
5	3, 4	ax-mp 5	1 ⊢ ((𝑅 hereditary 𝐴 → 𝑋 ∈ 𝐴) → (𝑅 hereditary 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)))

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146   class class class wbr 5111   hereditary whe 44576

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406  ax-frege1 44594  ax-frege2 44595  ax-frege8 44613  ax-frege52a 44661  ax-frege58b 44705

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 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-he 44577

This theorem is used by:  frege87  44754