Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj581 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj580 32295. 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). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) Remove unnecessary distinct variable conditions. (Revised by Andrew Salmon, 9-Jul-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 3776 | . 2 ⊢ ([𝑔 / 𝑓]𝜒 ↔ [𝑔 / 𝑓](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
4 | sbc3an 3785 | . . 3 ⊢ ([𝑔 / 𝑓](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ([𝑔 / 𝑓]𝑓 Fn 𝑛 ∧ [𝑔 / 𝑓]𝜑 ∧ [𝑔 / 𝑓]𝜓)) | |
5 | bnj62 32100 | . . . . 5 ⊢ ([𝑔 / 𝑓]𝑓 Fn 𝑛 ↔ 𝑔 Fn 𝑛) | |
6 | 5 | bicomi 227 | . . . 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 1152 | . . 3 ⊢ ((𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′) ↔ ([𝑔 / 𝑓]𝑓 Fn 𝑛 ∧ [𝑔 / 𝑓]𝜑 ∧ [𝑔 / 𝑓]𝜓)) |
10 | 4, 9 | bitr4i 281 | . 2 ⊢ ([𝑔 / 𝑓](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)) |
11 | 1, 3, 10 | 3bitri 300 | 1 ⊢ (𝜒′ ↔ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ w3a 1084 [wsbc 3720 Fn wfn 6319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-sbc 3721 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-fun 6326 df-fn 6327 |
This theorem is referenced by: bnj580 32295 bnj849 32307 |
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