Theorem nffvd 6657

Description: Deduction version of bound-variable hypothesis builder nffv 6655. (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 2963	. . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹}
2		nfaba1 2963	. . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}
3	1, 2	nffv 6655	. 2 ⊢ Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹}‘{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴})
4		nffvd.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐹)
5		nffvd.3	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴)
6		nfnfc1 2958	. . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐹
7		nfnfc1 2958	. . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴
8	6, 7	nfan 1900	. . . 4 ⊢ Ⅎ𝑥(Ⅎ𝑥𝐹 ∧ Ⅎ𝑥𝐴)
9		abidnf 3642	. . . . . 6 ⊢ (Ⅎ𝑥𝐹 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹} = 𝐹)
10	9	adantr 484	. . . . 5 ⊢ ((Ⅎ𝑥𝐹 ∧ Ⅎ𝑥𝐴) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹} = 𝐹)
11		abidnf 3642	. . . . . 6 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴)
12	11	adantl 485	. . . . 5 ⊢ ((Ⅎ𝑥𝐹 ∧ Ⅎ𝑥𝐴) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴)
13	10, 12	fveq12d 6652	. . . 4 ⊢ ((Ⅎ𝑥𝐹 ∧ Ⅎ𝑥𝐴) → ({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹}‘{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}) = (𝐹‘𝐴))
14	8, 13	nfceqdf 2951	. . 3 ⊢ ((Ⅎ𝑥𝐹 ∧ Ⅎ𝑥𝐴) → (Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹}‘{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}) ↔ Ⅎ𝑥(𝐹‘𝐴)))
15	4, 5, 14	syl2anc 587	. 2 ⊢ (𝜑 → (Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐹}‘{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}) ↔ Ⅎ𝑥(𝐹‘𝐴)))
16	3, 15	mpbii 236	1 ⊢ (𝜑 → Ⅎ𝑥(𝐹‘𝐴))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2111  {cab 2776  Ⅎwnfc 2936  ‘cfv 6324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332

This theorem is referenced by:  nfovd  7164  nfixpw  8463  nfixp  8464