Theorem bnj1534 35250

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 2924	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfcv 2924	. . 3 ⊢ Ⅎ𝑧𝐴
4		nfv 1943	. . 3 ⊢ Ⅎ𝑧(𝐹‘𝑥) ≠ (𝐻‘𝑥)
5		bnj1534.2	. . . . . 6 ⊢ (𝑤 ∈ 𝐹 → ∀𝑥 𝑤 ∈ 𝐹)
6	5	nfcii 2913	. . . . 5 ⊢ Ⅎ𝑥𝐹
7		nfcv 2924	. . . . 5 ⊢ Ⅎ𝑥𝑧
8	6, 7	nffv 6891	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧)
9		nfcv 2924	. . . 4 ⊢ Ⅎ𝑥(𝐻‘𝑧)
10	8, 9	nfne 3060	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝑧) ≠ (𝐻‘𝑧)
11		fveq2 6881	. . . 4 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
12		fveq2 6881	. . . 4 ⊢ (𝑥 = 𝑧 → (𝐻‘𝑥) = (𝐻‘𝑧))
13	11, 12	neeq12d 3018	. . 3 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) ≠ (𝐻‘𝑥) ↔ (𝐹‘𝑧) ≠ (𝐻‘𝑧)))
14	2, 3, 4, 10, 13	cbvrabw 3450	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐻‘𝑥)} = {𝑧 ∈ 𝐴 ∣ (𝐹‘𝑧) ≠ (𝐻‘𝑧)}
15	1, 14	eqtri 2785	1 ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ (𝐹‘𝑧) ≠ (𝐻‘𝑧)}

This proof depends on syntax axioms:   → wi 4  ∀wal 1567   = wceq 1569   ∈ wcel 2142   ≠ wne 2957  {crab 3415  ‘cfv 6536

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-iota 6492  df-fv 6544

This theorem is used by:  bnj1523  35468