Theorem nffvd 5498

Description: Deduction version of bound-variable hypothesis builder nffv 5496. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
nffvd.2	⊢ (𝜑 → Ⅎ𝑥𝐹)
nffvd.3	⊢ (𝜑 → Ⅎ𝑥𝐴)

Assertion

Ref	Expression
nffvd	⊢ (𝜑 → Ⅎ𝑥(𝐹‘𝐴))

Proof of Theorem nffvd

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfaba1 2314	. . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹}
2		nfaba1 2314	. . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}
3	1, 2	nffv 5496	. 2 ⊢ Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹}‘{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴})
4		nffvd.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐹)
5		nffvd.3	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴)
6		nfnfc1 2311	. . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐹
7		nfnfc1 2311	. . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴
8	6, 7	nfan 1553	. . . 4 ⊢ Ⅎ𝑥(Ⅎ𝑥𝐹 ∧ Ⅎ𝑥𝐴)
9		abidnf 2894	. . . . . 6 ⊢ (Ⅎ𝑥𝐹 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹} = 𝐹)
10	9	adantr 274	. . . . 5 ⊢ ((Ⅎ𝑥𝐹 ∧ Ⅎ𝑥𝐴) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹} = 𝐹)
11		abidnf 2894	. . . . . 6 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴)
12	11	adantl 275	. . . . 5 ⊢ ((Ⅎ𝑥𝐹 ∧ Ⅎ𝑥𝐴) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴)
13	10, 12	fveq12d 5493	. . . 4 ⊢ ((Ⅎ𝑥𝐹 ∧ Ⅎ𝑥𝐴) → ({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹}‘{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}) = (𝐹‘𝐴))
14	8, 13	nfceqdf 2307	. . 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))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1341   = wceq 1343   ∈ wcel 2136  {cab 2151  Ⅎwnfc 2295  ‘cfv 5188

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-iota 5153  df-fv 5196

This theorem is referenced by:  nfovd  5871  nfixpxy  6683