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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 3763 | . . 3 ⊢ ([𝑔 / 𝑓]𝜁0 ↔ [𝑔 / 𝑓](𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) |
4 | sbc3an 3772 | . . . 4 ⊢ ([𝑔 / 𝑓](𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) ↔ ([𝑔 / 𝑓]𝑓 Fn 1o ∧ [𝑔 / 𝑓]𝜑′ ∧ [𝑔 / 𝑓]𝜓′)) | |
5 | bnj62 31603 | . . . . 5 ⊢ ([𝑔 / 𝑓]𝑓 Fn 1o ↔ 𝑔 Fn 1o) | |
6 | bnj156.3 | . . . . . 6 ⊢ (𝜑1 ↔ [𝑔 / 𝑓]𝜑′) | |
7 | 6 | bicomi 225 | . . . . 5 ⊢ ([𝑔 / 𝑓]𝜑′ ↔ 𝜑1) |
8 | bnj156.4 | . . . . . 6 ⊢ (𝜓1 ↔ [𝑔 / 𝑓]𝜓′) | |
9 | 8 | bicomi 225 | . . . . 5 ⊢ ([𝑔 / 𝑓]𝜓′ ↔ 𝜓1) |
10 | 5, 7, 9 | 3anbi123i 1148 | . . . 4 ⊢ (([𝑔 / 𝑓]𝑓 Fn 1o ∧ [𝑔 / 𝑓]𝜑′ ∧ [𝑔 / 𝑓]𝜓′) ↔ (𝑔 Fn 1o ∧ 𝜑1 ∧ 𝜓1)) |
11 | 4, 10 | bitri 276 | . . 3 ⊢ ([𝑔 / 𝑓](𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) ↔ (𝑔 Fn 1o ∧ 𝜑1 ∧ 𝜓1)) |
12 | 3, 11 | bitri 276 | . 2 ⊢ ([𝑔 / 𝑓]𝜁0 ↔ (𝑔 Fn 1o ∧ 𝜑1 ∧ 𝜓1)) |
13 | 1, 12 | bitri 276 | 1 ⊢ (𝜁1 ↔ (𝑔 Fn 1o ∧ 𝜑1 ∧ 𝜓1)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ w3a 1080 [wsbc 3711 Fn wfn 6227 1oc1o 7953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-rab 3116 df-v 3442 df-sbc 3712 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-sn 4479 df-pr 4481 df-op 4485 df-br 4969 df-opab 5031 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-fun 6234 df-fn 6235 |
This theorem is referenced by: bnj153 31764 |
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