Theorem bnj1534 34615

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1534.1	⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐻‘𝑥)}
bnj1534.2	⊢ (𝑤 ∈ 𝐹 → ∀𝑥 𝑤 ∈ 𝐹)

Assertion

Ref	Expression
bnj1534	⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ (𝐹‘𝑧) ≠ (𝐻‘𝑧)}

Distinct variable groups:   𝑤,𝐴,𝑥,𝑧   𝑤,𝐹,𝑧   𝑤,𝐻,𝑥,𝑧

Allowed substitution hints:   𝐷(𝑥,𝑧,𝑤)   𝐹(𝑥)

Proof of Theorem bnj1534

Step	Hyp	Ref	Expression
1		bnj1534.1	. 2 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐻‘𝑥)}
2		nfcv 2891	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfcv 2891	. . 3 ⊢ Ⅎ𝑧𝐴
4		nfv 1909	. . 3 ⊢ Ⅎ𝑧(𝐹‘𝑥) ≠ (𝐻‘𝑥)
5		bnj1534.2	. . . . . 6 ⊢ (𝑤 ∈ 𝐹 → ∀𝑥 𝑤 ∈ 𝐹)
6	5	nfcii 2879	. . . . 5 ⊢ Ⅎ𝑥𝐹
7		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑥𝑧
8	6, 7	nffv 6906	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧)
9		nfcv 2891	. . . 4 ⊢ Ⅎ𝑥(𝐻‘𝑧)
10	8, 9	nfne 3032	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝑧) ≠ (𝐻‘𝑧)
11		fveq2 6896	. . . 4 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
12		fveq2 6896	. . . 4 ⊢ (𝑥 = 𝑧 → (𝐻‘𝑥) = (𝐻‘𝑧))
13	11, 12	neeq12d 2991	. . 3 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) ≠ (𝐻‘𝑥) ↔ (𝐹‘𝑧) ≠ (𝐻‘𝑧)))
14	2, 3, 4, 10, 13	cbvrabw 3455	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐻‘𝑥)} = {𝑧 ∈ 𝐴 ∣ (𝐹‘𝑧) ≠ (𝐻‘𝑧)}
15	1, 14	eqtri 2753	1 ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ (𝐹‘𝑧) ≠ (𝐻‘𝑧)}

Syntax hints:   → wi 4  ∀wal 1531   = wceq 1533   ∈ wcel 2098   ≠ wne 2929  {crab 3418  ‘cfv 6549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-iota 6501  df-fv 6557

This theorem is referenced by:  bnj1523  34833