Theorem nfimad 4955

Description: Deduction version of bound-variable hypothesis builder nfima 4954. (Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
nfimad.2	⊢ (φ → ℲxA)
nfimad.3	⊢ (φ → ℲxB)

Assertion

Ref	Expression
nfimad	⊢ (φ → Ⅎx(A “ B))

Proof of Theorem nfimad

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfaba1 2495	. . 3 ⊢ Ⅎx{z ∣ ∀x z ∈ A}
2		nfaba1 2495	. . 3 ⊢ Ⅎx{z ∣ ∀x z ∈ B}
3	1, 2	nfima 4954	. 2 ⊢ Ⅎx({z ∣ ∀x z ∈ A} “ {z ∣ ∀x z ∈ B})
4		nfimad.2	. . 3 ⊢ (φ → ℲxA)
5		nfimad.3	. . 3 ⊢ (φ → ℲxB)
6		nfnfc1 2493	. . . . 5 ⊢ ℲxℲxA
7		nfnfc1 2493	. . . . 5 ⊢ ℲxℲxB
8	6, 7	nfan 1824	. . . 4 ⊢ Ⅎx(ℲxA ∧ ℲxB)
9		abidnf 3006	. . . . . 6 ⊢ (ℲxA → {z ∣ ∀x z ∈ A} = A)
10	9	imaeq1d 4942	. . . . 5 ⊢ (ℲxA → ({z ∣ ∀x z ∈ A} “ {z ∣ ∀x z ∈ B}) = (A “ {z ∣ ∀x z ∈ B}))
11		abidnf 3006	. . . . . 6 ⊢ (ℲxB → {z ∣ ∀x z ∈ B} = B)
12	11	imaeq2d 4943	. . . . 5 ⊢ (ℲxB → (A “ {z ∣ ∀x z ∈ B}) = (A “ B))
13	10, 12	sylan9eq 2405	. . . 4 ⊢ ((ℲxA ∧ ℲxB) → ({z ∣ ∀x z ∈ A} “ {z ∣ ∀x z ∈ B}) = (A “ B))
14	8, 13	nfceqdf 2489	. . 3 ⊢ ((ℲxA ∧ ℲxB) → (Ⅎx({z ∣ ∀x z ∈ A} “ {z ∣ ∀x z ∈ B}) ↔ Ⅎx(A “ B)))
15	4, 5, 14	syl2anc 642	. 2 ⊢ (φ → (Ⅎx({z ∣ ∀x z ∈ A} “ {z ∣ ∀x z ∈ B}) ↔ Ⅎx(A “ B)))
16	3, 15	mpbii 202	1 ⊢ (φ → Ⅎx(A “ B))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2477   “ cima 4723

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-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-br 4641  df-ima 4728

This theorem is referenced by: (None)