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| Description: Lemma for frege124 44005. Proposition 123 of [Frege1879] p. 79. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.) | 
| Ref | Expression | 
|---|---|
| frege123.x | ⊢ 𝑋 ∈ 𝑈 | 
| frege123.y | ⊢ 𝑌 ∈ 𝑉 | 
| Ref | Expression | 
|---|---|
| frege123 | ⊢ ((∀𝑎(𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎) → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀)) → (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀)))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | frege123.x | . . . 4 ⊢ 𝑋 ∈ 𝑈 | |
| 2 | frege123.y | . . . 4 ⊢ 𝑌 ∈ 𝑉 | |
| 3 | vex 3483 | . . . 4 ⊢ 𝑎 ∈ V | |
| 4 | 1, 2, 3 | frege122 44003 | . . 3 ⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎))) | 
| 5 | 4 | alrimdv 1928 | . 2 ⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎))) | 
| 6 | frege19 43842 | . 2 ⊢ ((Fun ◡◡𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎))) → ((∀𝑎(𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎) → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀)) → (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀))))) | |
| 7 | 5, 6 | ax-mp 5 | 1 ⊢ ((∀𝑎(𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎) → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀)) → (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀)))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1537 ∈ wcel 2107 Vcvv 3479 ∪ cun 3948 class class class wbr 5142 I cid 5576 ◡ccnv 5683 Fun wfun 6554 ‘cfv 6560 t+ctcl 15025 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-frege1 43808 ax-frege2 43809 ax-frege8 43827 ax-frege52a 43875 ax-frege58b 43919 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-fun 6562 | 
| This theorem is referenced by: frege124 44005 | 
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