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Mirrors > Home > MPE Home > Th. List > nffvd | Structured version Visualization version GIF version |
Description: Deduction version of bound-variable hypothesis builder nffv 6673. (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 2983 | . . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹} | |
2 | nfaba1 2983 | . . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} | |
3 | 1, 2 | nffv 6673 | . 2 ⊢ Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹}‘{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}) |
4 | nffvd.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐹) | |
5 | nffvd.3 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
6 | nfnfc1 2977 | . . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐹 | |
7 | nfnfc1 2977 | . . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴 | |
8 | 6, 7 | nfan 1891 | . . . 4 ⊢ Ⅎ𝑥(Ⅎ𝑥𝐹 ∧ Ⅎ𝑥𝐴) |
9 | abidnf 3691 | . . . . . 6 ⊢ (Ⅎ𝑥𝐹 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹} = 𝐹) | |
10 | 9 | adantr 481 | . . . . 5 ⊢ ((Ⅎ𝑥𝐹 ∧ Ⅎ𝑥𝐴) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹} = 𝐹) |
11 | abidnf 3691 | . . . . . 6 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴) | |
12 | 11 | adantl 482 | . . . . 5 ⊢ ((Ⅎ𝑥𝐹 ∧ Ⅎ𝑥𝐴) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴) |
13 | 10, 12 | fveq12d 6670 | . . . 4 ⊢ ((Ⅎ𝑥𝐹 ∧ Ⅎ𝑥𝐴) → ({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹}‘{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}) = (𝐹‘𝐴)) |
14 | 8, 13 | nfceqdf 2969 | . . 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 234 | 1 ⊢ (𝜑 → Ⅎ𝑥(𝐹‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∀wal 1526 = wceq 1528 ∈ wcel 2105 {cab 2796 Ⅎwnfc 2958 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 |
This theorem is referenced by: nfovd 7174 nfixpw 8468 nfixp 8469 |
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