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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege92 | Structured version Visualization version GIF version |
Description: Inference from frege91 39204. Proposition 92 of [Frege1879] p. 69. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege91.x | ⊢ 𝑋 ∈ 𝑈 |
frege91.y | ⊢ 𝑌 ∈ 𝑉 |
frege91.r | ⊢ 𝑅 ∈ 𝑊 |
Ref | Expression |
---|---|
frege92 | ⊢ (𝑋 = 𝑍 → (𝑋𝑅𝑌 → 𝑍(t+‘𝑅)𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege91.x | . 2 ⊢ 𝑋 ∈ 𝑈 | |
2 | vex 3401 | . . . . 5 ⊢ 𝑤 ∈ V | |
3 | frege91.y | . . . . 5 ⊢ 𝑌 ∈ 𝑉 | |
4 | frege91.r | . . . . 5 ⊢ 𝑅 ∈ 𝑊 | |
5 | 2, 3, 4 | frege91 39204 | . . . 4 ⊢ (𝑤𝑅𝑌 → 𝑤(t+‘𝑅)𝑌) |
6 | 5 | sbcth 3667 | . . 3 ⊢ (𝑋 ∈ 𝑈 → [𝑋 / 𝑤](𝑤𝑅𝑌 → 𝑤(t+‘𝑅)𝑌)) |
7 | frege53c 39164 | . . 3 ⊢ ([𝑋 / 𝑤](𝑤𝑅𝑌 → 𝑤(t+‘𝑅)𝑌) → (𝑋 = 𝑍 → [𝑍 / 𝑤](𝑤𝑅𝑌 → 𝑤(t+‘𝑅)𝑌))) | |
8 | 6, 7 | syl 17 | . 2 ⊢ (𝑋 ∈ 𝑈 → (𝑋 = 𝑍 → [𝑍 / 𝑤](𝑤𝑅𝑌 → 𝑤(t+‘𝑅)𝑌))) |
9 | sbcim1 3700 | . . . 4 ⊢ ([𝑍 / 𝑤](𝑤𝑅𝑌 → 𝑤(t+‘𝑅)𝑌) → ([𝑍 / 𝑤]𝑤𝑅𝑌 → [𝑍 / 𝑤]𝑤(t+‘𝑅)𝑌)) | |
10 | 9 | imim2i 16 | . . 3 ⊢ ((𝑋 = 𝑍 → [𝑍 / 𝑤](𝑤𝑅𝑌 → 𝑤(t+‘𝑅)𝑌)) → (𝑋 = 𝑍 → ([𝑍 / 𝑤]𝑤𝑅𝑌 → [𝑍 / 𝑤]𝑤(t+‘𝑅)𝑌))) |
11 | sbcbr1g 4943 | . . . . . . 7 ⊢ (𝑋 ∈ 𝑈 → ([𝑋 / 𝑤]𝑤𝑅𝑌 ↔ ⦋𝑋 / 𝑤⦌𝑤𝑅𝑌)) | |
12 | csbvarg 4228 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝑈 → ⦋𝑋 / 𝑤⦌𝑤 = 𝑋) | |
13 | 12 | breq1d 4896 | . . . . . . 7 ⊢ (𝑋 ∈ 𝑈 → (⦋𝑋 / 𝑤⦌𝑤𝑅𝑌 ↔ 𝑋𝑅𝑌)) |
14 | 11, 13 | bitrd 271 | . . . . . 6 ⊢ (𝑋 ∈ 𝑈 → ([𝑋 / 𝑤]𝑤𝑅𝑌 ↔ 𝑋𝑅𝑌)) |
15 | 1, 14 | ax-mp 5 | . . . . 5 ⊢ ([𝑋 / 𝑤]𝑤𝑅𝑌 ↔ 𝑋𝑅𝑌) |
16 | dfsbcq 3654 | . . . . 5 ⊢ (𝑋 = 𝑍 → ([𝑋 / 𝑤]𝑤𝑅𝑌 ↔ [𝑍 / 𝑤]𝑤𝑅𝑌)) | |
17 | 15, 16 | syl5rbbr 278 | . . . 4 ⊢ (𝑋 = 𝑍 → ([𝑍 / 𝑤]𝑤𝑅𝑌 ↔ 𝑋𝑅𝑌)) |
18 | eqcom 2785 | . . . . . . 7 ⊢ (𝑋 = 𝑍 ↔ 𝑍 = 𝑋) | |
19 | 18 | biimpi 208 | . . . . . 6 ⊢ (𝑋 = 𝑍 → 𝑍 = 𝑋) |
20 | 19, 1 | syl6eqel 2867 | . . . . 5 ⊢ (𝑋 = 𝑍 → 𝑍 ∈ 𝑈) |
21 | sbcbr1g 4943 | . . . . . 6 ⊢ (𝑍 ∈ 𝑈 → ([𝑍 / 𝑤]𝑤(t+‘𝑅)𝑌 ↔ ⦋𝑍 / 𝑤⦌𝑤(t+‘𝑅)𝑌)) | |
22 | csbvarg 4228 | . . . . . . 7 ⊢ (𝑍 ∈ 𝑈 → ⦋𝑍 / 𝑤⦌𝑤 = 𝑍) | |
23 | 22 | breq1d 4896 | . . . . . 6 ⊢ (𝑍 ∈ 𝑈 → (⦋𝑍 / 𝑤⦌𝑤(t+‘𝑅)𝑌 ↔ 𝑍(t+‘𝑅)𝑌)) |
24 | 21, 23 | bitrd 271 | . . . . 5 ⊢ (𝑍 ∈ 𝑈 → ([𝑍 / 𝑤]𝑤(t+‘𝑅)𝑌 ↔ 𝑍(t+‘𝑅)𝑌)) |
25 | 20, 24 | syl 17 | . . . 4 ⊢ (𝑋 = 𝑍 → ([𝑍 / 𝑤]𝑤(t+‘𝑅)𝑌 ↔ 𝑍(t+‘𝑅)𝑌)) |
26 | 17, 25 | imbi12d 336 | . . 3 ⊢ (𝑋 = 𝑍 → (([𝑍 / 𝑤]𝑤𝑅𝑌 → [𝑍 / 𝑤]𝑤(t+‘𝑅)𝑌) ↔ (𝑋𝑅𝑌 → 𝑍(t+‘𝑅)𝑌))) |
27 | 10, 26 | mpbidi 233 | . 2 ⊢ ((𝑋 = 𝑍 → [𝑍 / 𝑤](𝑤𝑅𝑌 → 𝑤(t+‘𝑅)𝑌)) → (𝑋 = 𝑍 → (𝑋𝑅𝑌 → 𝑍(t+‘𝑅)𝑌))) |
28 | 1, 8, 27 | mp2b 10 | 1 ⊢ (𝑋 = 𝑍 → (𝑋𝑅𝑌 → 𝑍(t+‘𝑅)𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 ∈ wcel 2107 Vcvv 3398 [wsbc 3652 ⦋csb 3751 class class class wbr 4886 ‘cfv 6135 t+ctcl 14133 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-frege1 39040 ax-frege2 39041 ax-frege8 39059 ax-frege52a 39107 ax-frege52c 39138 ax-frege58b 39151 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-ifp 1047 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-n0 11643 df-z 11729 df-uz 11993 df-seq 13120 df-trcl 14135 df-relexp 14168 df-he 39023 |
This theorem is referenced by: frege102 39215 |
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