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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege73 | Structured version Visualization version GIF version |
Description: Lemma for frege87 43853. Proposition 73 of [Frege1879] p. 59. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege73.x | ⊢ 𝑋 ∈ 𝑈 |
frege73.y | ⊢ 𝑌 ∈ 𝑉 |
Ref | Expression |
---|---|
frege73 | ⊢ ((𝑅 hereditary 𝐴 → 𝑋 ∈ 𝐴) → (𝑅 hereditary 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege73.x | . . 3 ⊢ 𝑋 ∈ 𝑈 | |
2 | frege73.y | . . 3 ⊢ 𝑌 ∈ 𝑉 | |
3 | 1, 2 | frege72 43838 | . 2 ⊢ (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴))) |
4 | ax-frege2 43694 | . 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))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2103 class class class wbr 5169 hereditary whe 43675 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pr 5450 ax-frege1 43693 ax-frege2 43694 ax-frege8 43712 ax-frege52a 43760 ax-frege58b 43804 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-ifp 1064 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-br 5170 df-opab 5232 df-xp 5705 df-cnv 5707 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-he 43676 |
This theorem is referenced by: frege87 43853 |
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