Theorem bnj1534 32235

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 2955	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfcv 2955	. . 3 ⊢ Ⅎ𝑧𝐴
4		nfv 1915	. . 3 ⊢ Ⅎ𝑧(𝐹‘𝑥) ≠ (𝐻‘𝑥)
5		bnj1534.2	. . . . . 6 ⊢ (𝑤 ∈ 𝐹 → ∀𝑥 𝑤 ∈ 𝐹)
6	5	nfcii 2940	. . . . 5 ⊢ Ⅎ𝑥𝐹
7		nfcv 2955	. . . . 5 ⊢ Ⅎ𝑥𝑧
8	6, 7	nffv 6655	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧)
9		nfcv 2955	. . . 4 ⊢ Ⅎ𝑥(𝐻‘𝑧)
10	8, 9	nfne 3087	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝑧) ≠ (𝐻‘𝑧)
11		fveq2 6645	. . . 4 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
12		fveq2 6645	. . . 4 ⊢ (𝑥 = 𝑧 → (𝐻‘𝑥) = (𝐻‘𝑧))
13	11, 12	neeq12d 3048	. . 3 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) ≠ (𝐻‘𝑥) ↔ (𝐹‘𝑧) ≠ (𝐻‘𝑧)))
14	2, 3, 4, 10, 13	cbvrabw 3437	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐻‘𝑥)} = {𝑧 ∈ 𝐴 ∣ (𝐹‘𝑧) ≠ (𝐻‘𝑧)}
15	1, 14	eqtri 2821	1 ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ (𝐹‘𝑧) ≠ (𝐻‘𝑧)}

Syntax hints:   → wi 4  ∀wal 1536   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  {crab 3110  ‘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-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  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:  bnj1523  32453