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Mirrors > Home > ILE Home > Th. List > nffvd | GIF version |
Description: Deduction version of bound-variable hypothesis builder nffv 5439. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Ref | Expression |
---|---|
nffvd.2 | ⊢ (𝜑 → Ⅎ𝑥𝐹) |
nffvd.3 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
Ref | Expression |
---|---|
nffvd | ⊢ (𝜑 → Ⅎ𝑥(𝐹‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfaba1 2288 | . . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹} | |
2 | nfaba1 2288 | . . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} | |
3 | 1, 2 | nffv 5439 | . 2 ⊢ Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹}‘{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}) |
4 | nffvd.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐹) | |
5 | nffvd.3 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
6 | nfnfc1 2285 | . . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐹 | |
7 | nfnfc1 2285 | . . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴 | |
8 | 6, 7 | nfan 1545 | . . . 4 ⊢ Ⅎ𝑥(Ⅎ𝑥𝐹 ∧ Ⅎ𝑥𝐴) |
9 | abidnf 2856 | . . . . . 6 ⊢ (Ⅎ𝑥𝐹 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹} = 𝐹) | |
10 | 9 | adantr 274 | . . . . 5 ⊢ ((Ⅎ𝑥𝐹 ∧ Ⅎ𝑥𝐴) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹} = 𝐹) |
11 | abidnf 2856 | . . . . . 6 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴) | |
12 | 11 | adantl 275 | . . . . 5 ⊢ ((Ⅎ𝑥𝐹 ∧ Ⅎ𝑥𝐴) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴) |
13 | 10, 12 | fveq12d 5436 | . . . 4 ⊢ ((Ⅎ𝑥𝐹 ∧ Ⅎ𝑥𝐴) → ({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹}‘{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}) = (𝐹‘𝐴)) |
14 | 8, 13 | nfceqdf 2281 | . . 3 ⊢ ((Ⅎ𝑥𝐹 ∧ Ⅎ𝑥𝐴) → (Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹}‘{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}) ↔ Ⅎ𝑥(𝐹‘𝐴))) |
15 | 4, 5, 14 | syl2anc 409 | . 2 ⊢ (𝜑 → (Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹}‘{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}) ↔ Ⅎ𝑥(𝐹‘𝐴))) |
16 | 3, 15 | mpbii 147 | 1 ⊢ (𝜑 → Ⅎ𝑥(𝐹‘𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1330 = wceq 1332 ∈ wcel 1481 {cab 2126 Ⅎwnfc 2269 ‘cfv 5131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-rex 2423 df-v 2691 df-un 3080 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-iota 5096 df-fv 5139 |
This theorem is referenced by: nfovd 5808 nfixpxy 6619 |
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