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 30983. 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 3491 | . 2 ⊢ ([𝑔 / 𝑓]𝜒 ↔ [𝑔 / 𝑓](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
4 | sbc3an 3494 | . . 3 ⊢ ([𝑔 / 𝑓](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ([𝑔 / 𝑓]𝑓 Fn 𝑛 ∧ [𝑔 / 𝑓]𝜑 ∧ [𝑔 / 𝑓]𝜓)) | |
5 | bnj62 30786 | . . . . 5 ⊢ ([𝑔 / 𝑓]𝑓 Fn 𝑛 ↔ 𝑔 Fn 𝑛) | |
6 | 5 | bicomi 214 | . . . 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 1251 | . . 3 ⊢ ((𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′) ↔ ([𝑔 / 𝑓]𝑓 Fn 𝑛 ∧ [𝑔 / 𝑓]𝜑 ∧ [𝑔 / 𝑓]𝜓)) |
10 | 4, 9 | bitr4i 267 | . 2 ⊢ ([𝑔 / 𝑓](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)) |
11 | 1, 3, 10 | 3bitri 286 | 1 ⊢ (𝜒′ ↔ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ w3a 1037 [wsbc 3435 Fn wfn 5883 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-fun 5890 df-fn 5891 |
This theorem is referenced by: bnj580 30983 bnj849 30995 |
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