![]() |
Mathbox for Jonathan Ben-Naim |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj581 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj580 31500. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Unnecessary distinct variable restrictions were removed by Andrew Salmon, 9-Jul-2011.) (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj581.3 | ⊢ (𝜒 ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
bnj581.4 | ⊢ (𝜑′ ↔ [𝑔 / 𝑓]𝜑) |
bnj581.5 | ⊢ (𝜓′ ↔ [𝑔 / 𝑓]𝜓) |
bnj581.6 | ⊢ (𝜒′ ↔ [𝑔 / 𝑓]𝜒) |
Ref | Expression |
---|---|
bnj581 | ⊢ (𝜒′ ↔ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj581.6 | . 2 ⊢ (𝜒′ ↔ [𝑔 / 𝑓]𝜒) | |
2 | bnj581.3 | . . 3 ⊢ (𝜒 ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) | |
3 | 2 | sbcbii 3689 | . 2 ⊢ ([𝑔 / 𝑓]𝜒 ↔ [𝑔 / 𝑓](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
4 | sbc3an 3691 | . . 3 ⊢ ([𝑔 / 𝑓](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ([𝑔 / 𝑓]𝑓 Fn 𝑛 ∧ [𝑔 / 𝑓]𝜑 ∧ [𝑔 / 𝑓]𝜓)) | |
5 | bnj62 31306 | . . . . 5 ⊢ ([𝑔 / 𝑓]𝑓 Fn 𝑛 ↔ 𝑔 Fn 𝑛) | |
6 | 5 | bicomi 216 | . . . 4 ⊢ (𝑔 Fn 𝑛 ↔ [𝑔 / 𝑓]𝑓 Fn 𝑛) |
7 | bnj581.4 | . . . 4 ⊢ (𝜑′ ↔ [𝑔 / 𝑓]𝜑) | |
8 | bnj581.5 | . . . 4 ⊢ (𝜓′ ↔ [𝑔 / 𝑓]𝜓) | |
9 | 6, 7, 8 | 3anbi123i 1195 | . . 3 ⊢ ((𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′) ↔ ([𝑔 / 𝑓]𝑓 Fn 𝑛 ∧ [𝑔 / 𝑓]𝜑 ∧ [𝑔 / 𝑓]𝜓)) |
10 | 4, 9 | bitr4i 270 | . 2 ⊢ ([𝑔 / 𝑓](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)) |
11 | 1, 3, 10 | 3bitri 289 | 1 ⊢ (𝜒′ ↔ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ w3a 1108 [wsbc 3633 Fn wfn 6096 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-br 4844 df-opab 4906 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-fun 6103 df-fn 6104 |
This theorem is referenced by: bnj580 31500 bnj849 31512 |
Copyright terms: Public domain | W3C validator |