Theorem nffvd 5336

Description: Deduction version of bound-variable hypothesis builder nffv 5335. (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 2495	. . 3 ⊢ Ⅎx{z ∣ ∀x z ∈ F}
2		nfaba1 2495	. . 3 ⊢ Ⅎx{z ∣ ∀x z ∈ A}
3	1, 2	nffv 5335	. 2 ⊢ Ⅎx({z ∣ ∀x z ∈ F} ‘{z ∣ ∀x z ∈ A})
4		nffvd.2	. . 3 ⊢ (φ → ℲxF)
5		nffvd.3	. . 3 ⊢ (φ → ℲxA)
6		nfnfc1 2493	. . . . 5 ⊢ ℲxℲxF
7		nfnfc1 2493	. . . . 5 ⊢ ℲxℲxA
8	6, 7	nfan 1824	. . . 4 ⊢ Ⅎx(ℲxF ∧ ℲxA)
9		abidnf 3006	. . . . . 6 ⊢ (ℲxF → {z ∣ ∀x z ∈ F} = F)
10	9	adantr 451	. . . . 5 ⊢ ((ℲxF ∧ ℲxA) → {z ∣ ∀x z ∈ F} = F)
11		abidnf 3006	. . . . . 6 ⊢ (ℲxA → {z ∣ ∀x z ∈ A} = A)
12	11	adantl 452	. . . . 5 ⊢ ((ℲxF ∧ ℲxA) → {z ∣ ∀x z ∈ A} = A)
13	10, 12	fveq12d 5334	. . . 4 ⊢ ((ℲxF ∧ ℲxA) → ({z ∣ ∀x z ∈ F} ‘{z ∣ ∀x z ∈ A}) = (F ‘A))
14	8, 13	nfceqdf 2489	. . 3 ⊢ ((ℲxF ∧ ℲxA) → (Ⅎx({z ∣ ∀x z ∈ F} ‘{z ∣ ∀x z ∈ A}) ↔ Ⅎx(F ‘A)))
15	4, 5, 14	syl2anc 642	. 2 ⊢ (φ → (Ⅎx({z ∣ ∀x z ∈ F} ‘{z ∣ ∀x z ∈ A}) ↔ Ⅎx(F ‘A)))
16	3, 15	mpbii 202	1 ⊢ (φ → Ⅎx(F ‘A))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2477   ‘cfv 4782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-br 4641  df-fv 4796

This theorem is referenced by:  nfovd  5545