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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 6110. (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 2756 | . . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹} | |
2 | nfaba1 2756 | . . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} | |
3 | 1, 2 | nffv 6110 | . 2 ⊢ Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹}‘{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}) |
4 | nffvd.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐹) | |
5 | nffvd.3 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
6 | nfnfc1 2754 | . . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐹 | |
7 | nfnfc1 2754 | . . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴 | |
8 | 6, 7 | nfan 1816 | . . . 4 ⊢ Ⅎ𝑥(Ⅎ𝑥𝐹 ∧ Ⅎ𝑥𝐴) |
9 | abidnf 3342 | . . . . . 6 ⊢ (Ⅎ𝑥𝐹 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹} = 𝐹) | |
10 | 9 | adantr 480 | . . . . 5 ⊢ ((Ⅎ𝑥𝐹 ∧ Ⅎ𝑥𝐴) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹} = 𝐹) |
11 | abidnf 3342 | . . . . . 6 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴) | |
12 | 11 | adantl 481 | . . . . 5 ⊢ ((Ⅎ𝑥𝐹 ∧ Ⅎ𝑥𝐴) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴) |
13 | 10, 12 | fveq12d 6109 | . . . 4 ⊢ ((Ⅎ𝑥𝐹 ∧ Ⅎ𝑥𝐴) → ({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹}‘{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}) = (𝐹‘𝐴)) |
14 | 8, 13 | nfceqdf 2747 | . . 3 ⊢ ((Ⅎ𝑥𝐹 ∧ Ⅎ𝑥𝐴) → (Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹}‘{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}) ↔ Ⅎ𝑥(𝐹‘𝐴))) |
15 | 4, 5, 14 | syl2anc 691 | . 2 ⊢ (𝜑 → (Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹}‘{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}) ↔ Ⅎ𝑥(𝐹‘𝐴))) |
16 | 3, 15 | mpbii 222 | 1 ⊢ (𝜑 → Ⅎ𝑥(𝐹‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∀wal 1473 = wceq 1475 ∈ wcel 1977 {cab 2596 Ⅎwnfc 2738 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 |
This theorem is referenced by: nfovd 6574 nfixp 7813 |
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