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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj156 | Structured version Visualization version GIF version |
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj156.1 | ⊢ (𝜁0 ↔ (𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′)) |
bnj156.2 | ⊢ (𝜁1 ↔ [𝑔 / 𝑓]𝜁0) |
bnj156.3 | ⊢ (𝜑1 ↔ [𝑔 / 𝑓]𝜑′) |
bnj156.4 | ⊢ (𝜓1 ↔ [𝑔 / 𝑓]𝜓′) |
Ref | Expression |
---|---|
bnj156 | ⊢ (𝜁1 ↔ (𝑔 Fn 1𝑜 ∧ 𝜑1 ∧ 𝜓1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj156.2 | . 2 ⊢ (𝜁1 ↔ [𝑔 / 𝑓]𝜁0) | |
2 | bnj156.1 | . . . 4 ⊢ (𝜁0 ↔ (𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′)) | |
3 | 2 | sbcbii 3689 | . . 3 ⊢ ([𝑔 / 𝑓]𝜁0 ↔ [𝑔 / 𝑓](𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′)) |
4 | sbc3an 3691 | . . . 4 ⊢ ([𝑔 / 𝑓](𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′) ↔ ([𝑔 / 𝑓]𝑓 Fn 1𝑜 ∧ [𝑔 / 𝑓]𝜑′ ∧ [𝑔 / 𝑓]𝜓′)) | |
5 | bnj62 31306 | . . . . 5 ⊢ ([𝑔 / 𝑓]𝑓 Fn 1𝑜 ↔ 𝑔 Fn 1𝑜) | |
6 | bnj156.3 | . . . . . 6 ⊢ (𝜑1 ↔ [𝑔 / 𝑓]𝜑′) | |
7 | 6 | bicomi 216 | . . . . 5 ⊢ ([𝑔 / 𝑓]𝜑′ ↔ 𝜑1) |
8 | bnj156.4 | . . . . . 6 ⊢ (𝜓1 ↔ [𝑔 / 𝑓]𝜓′) | |
9 | 8 | bicomi 216 | . . . . 5 ⊢ ([𝑔 / 𝑓]𝜓′ ↔ 𝜓1) |
10 | 5, 7, 9 | 3anbi123i 1195 | . . . 4 ⊢ (([𝑔 / 𝑓]𝑓 Fn 1𝑜 ∧ [𝑔 / 𝑓]𝜑′ ∧ [𝑔 / 𝑓]𝜓′) ↔ (𝑔 Fn 1𝑜 ∧ 𝜑1 ∧ 𝜓1)) |
11 | 4, 10 | bitri 267 | . . 3 ⊢ ([𝑔 / 𝑓](𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′) ↔ (𝑔 Fn 1𝑜 ∧ 𝜑1 ∧ 𝜓1)) |
12 | 3, 11 | bitri 267 | . 2 ⊢ ([𝑔 / 𝑓]𝜁0 ↔ (𝑔 Fn 1𝑜 ∧ 𝜑1 ∧ 𝜓1)) |
13 | 1, 12 | bitri 267 | 1 ⊢ (𝜁1 ↔ (𝑔 Fn 1𝑜 ∧ 𝜑1 ∧ 𝜓1)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ w3a 1108 [wsbc 3633 Fn wfn 6096 1𝑜c1o 7792 |
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: bnj153 31467 |
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