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| Mirrors > Home > MPE Home > Th. List > Mathboxes > frege120 | Structured version Visualization version GIF version | ||
| Description: Simplified application of one direction of dffrege115 43991. Proposition 120 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.) |
| Ref | Expression |
|---|---|
| frege116.x | ⊢ 𝑋 ∈ 𝑈 |
| frege118.y | ⊢ 𝑌 ∈ 𝑉 |
| frege120.a | ⊢ 𝐴 ∈ 𝑊 |
| Ref | Expression |
|---|---|
| frege120 | ⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝐴 = 𝑋))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | frege120.a | . . . 4 ⊢ 𝐴 ∈ 𝑊 | |
| 2 | 1 | frege58c 43934 | . . 3 ⊢ (∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋) → [𝐴 / 𝑎](𝑌𝑅𝑎 → 𝑎 = 𝑋)) |
| 3 | sbcim1 3842 | . . . 4 ⊢ ([𝐴 / 𝑎](𝑌𝑅𝑎 → 𝑎 = 𝑋) → ([𝐴 / 𝑎]𝑌𝑅𝑎 → [𝐴 / 𝑎]𝑎 = 𝑋)) | |
| 4 | sbcbr2g 5201 | . . . . . 6 ⊢ (𝐴 ∈ 𝑊 → ([𝐴 / 𝑎]𝑌𝑅𝑎 ↔ 𝑌𝑅⦋𝐴 / 𝑎⦌𝑎)) | |
| 5 | 1, 4 | ax-mp 5 | . . . . 5 ⊢ ([𝐴 / 𝑎]𝑌𝑅𝑎 ↔ 𝑌𝑅⦋𝐴 / 𝑎⦌𝑎) |
| 6 | csbvarg 4434 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑊 → ⦋𝐴 / 𝑎⦌𝑎 = 𝐴) | |
| 7 | 1, 6 | ax-mp 5 | . . . . . 6 ⊢ ⦋𝐴 / 𝑎⦌𝑎 = 𝐴 |
| 8 | 7 | breq2i 5151 | . . . . 5 ⊢ (𝑌𝑅⦋𝐴 / 𝑎⦌𝑎 ↔ 𝑌𝑅𝐴) |
| 9 | 5, 8 | bitri 275 | . . . 4 ⊢ ([𝐴 / 𝑎]𝑌𝑅𝑎 ↔ 𝑌𝑅𝐴) |
| 10 | sbceq1g 4417 | . . . . . 6 ⊢ (𝐴 ∈ 𝑊 → ([𝐴 / 𝑎]𝑎 = 𝑋 ↔ ⦋𝐴 / 𝑎⦌𝑎 = 𝑋)) | |
| 11 | 1, 10 | ax-mp 5 | . . . . 5 ⊢ ([𝐴 / 𝑎]𝑎 = 𝑋 ↔ ⦋𝐴 / 𝑎⦌𝑎 = 𝑋) |
| 12 | 7 | eqeq1i 2742 | . . . . 5 ⊢ (⦋𝐴 / 𝑎⦌𝑎 = 𝑋 ↔ 𝐴 = 𝑋) |
| 13 | 11, 12 | bitri 275 | . . . 4 ⊢ ([𝐴 / 𝑎]𝑎 = 𝑋 ↔ 𝐴 = 𝑋) |
| 14 | 3, 9, 13 | 3imtr3g 295 | . . 3 ⊢ ([𝐴 / 𝑎](𝑌𝑅𝑎 → 𝑎 = 𝑋) → (𝑌𝑅𝐴 → 𝐴 = 𝑋)) |
| 15 | 2, 14 | syl 17 | . 2 ⊢ (∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋) → (𝑌𝑅𝐴 → 𝐴 = 𝑋)) |
| 16 | frege116.x | . . 3 ⊢ 𝑋 ∈ 𝑈 | |
| 17 | frege118.y | . . 3 ⊢ 𝑌 ∈ 𝑉 | |
| 18 | 16, 17 | frege119 43995 | . 2 ⊢ ((∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋) → (𝑌𝑅𝐴 → 𝐴 = 𝑋)) → (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝐴 = 𝑋)))) |
| 19 | 15, 18 | ax-mp 5 | 1 ⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝐴 = 𝑋))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 = wceq 1540 ∈ wcel 2108 [wsbc 3788 ⦋csb 3899 class class class wbr 5143 ◡ccnv 5684 Fun wfun 6555 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-frege1 43803 ax-frege2 43804 ax-frege8 43822 ax-frege52a 43870 ax-frege58b 43914 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ifp 1064 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-fun 6563 |
| This theorem is referenced by: frege121 43997 |
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