![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nffvd | Structured version Visualization version GIF version |
Description: Deduction version of bound-variable hypothesis builder nffv 6917. (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 2911 | . . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹} | |
2 | nfaba1 2911 | . . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} | |
3 | 1, 2 | nffv 6917 | . 2 ⊢ Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹}‘{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}) |
4 | nffvd.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐹) | |
5 | nffvd.3 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
6 | nfnfc1 2906 | . . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐹 | |
7 | nfnfc1 2906 | . . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴 | |
8 | 6, 7 | nfan 1897 | . . . 4 ⊢ Ⅎ𝑥(Ⅎ𝑥𝐹 ∧ Ⅎ𝑥𝐴) |
9 | abidnf 3711 | . . . . . 6 ⊢ (Ⅎ𝑥𝐹 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹} = 𝐹) | |
10 | 9 | adantr 480 | . . . . 5 ⊢ ((Ⅎ𝑥𝐹 ∧ Ⅎ𝑥𝐴) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹} = 𝐹) |
11 | abidnf 3711 | . . . . . 6 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴) | |
12 | 11 | adantl 481 | . . . . 5 ⊢ ((Ⅎ𝑥𝐹 ∧ Ⅎ𝑥𝐴) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴) |
13 | 10, 12 | fveq12d 6914 | . . . 4 ⊢ ((Ⅎ𝑥𝐹 ∧ Ⅎ𝑥𝐴) → ({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹}‘{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}) = (𝐹‘𝐴)) |
14 | 8, 13 | nfceqdf 2899 | . . 3 ⊢ ((Ⅎ𝑥𝐹 ∧ Ⅎ𝑥𝐴) → (Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹}‘{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}) ↔ Ⅎ𝑥(𝐹‘𝐴))) |
15 | 4, 5, 14 | syl2anc 584 | . 2 ⊢ (𝜑 → (Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹}‘{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}) ↔ Ⅎ𝑥(𝐹‘𝐴))) |
16 | 3, 15 | mpbii 233 | 1 ⊢ (𝜑 → Ⅎ𝑥(𝐹‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1535 = wceq 1537 ∈ wcel 2106 {cab 2712 Ⅎwnfc 2888 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 |
This theorem is referenced by: nfovd 7460 nfixpw 8955 nfixp 8956 bj-gabima 36923 |
Copyright terms: Public domain | W3C validator |