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 32893. 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 3786 | . . 3 ⊢ ([𝑔 / 𝑓](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ([𝑔 / 𝑓]𝑓 Fn 𝑛 ∧ [𝑔 / 𝑓]𝜑 ∧ [𝑔 / 𝑓]𝜓)) | |
5 | bnj62 32699 | . . . . 5 ⊢ ([𝑔 / 𝑓]𝑓 Fn 𝑛 ↔ 𝑔 Fn 𝑛) | |
6 | 5 | bicomi 223 | . . . 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 1154 | . . 3 ⊢ ((𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′) ↔ ([𝑔 / 𝑓]𝑓 Fn 𝑛 ∧ [𝑔 / 𝑓]𝜑 ∧ [𝑔 / 𝑓]𝜓)) |
10 | 4, 9 | bitr4i 277 | . 2 ⊢ ([𝑔 / 𝑓](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)) |
11 | 1, 3, 10 | 3bitri 297 | 1 ⊢ (𝜒′ ↔ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ w3a 1086 [wsbc 3716 Fn wfn 6428 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 df-opab 5137 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-fun 6435 df-fn 6436 |
This theorem is referenced by: bnj580 32893 bnj849 32905 |
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