Theorem bnj1534 34988

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 2897	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfcv 2897	. . 3 ⊢ Ⅎ𝑧𝐴
4		nfv 1916	. . 3 ⊢ Ⅎ𝑧(𝐹‘𝑥) ≠ (𝐻‘𝑥)
5		bnj1534.2	. . . . . 6 ⊢ (𝑤 ∈ 𝐹 → ∀𝑥 𝑤 ∈ 𝐹)
6	5	nfcii 2886	. . . . 5 ⊢ Ⅎ𝑥𝐹
7		nfcv 2897	. . . . 5 ⊢ Ⅎ𝑥𝑧
8	6, 7	nffv 6843	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧)
9		nfcv 2897	. . . 4 ⊢ Ⅎ𝑥(𝐻‘𝑧)
10	8, 9	nfne 3032	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝑧) ≠ (𝐻‘𝑧)
11		fveq2 6833	. . . 4 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
12		fveq2 6833	. . . 4 ⊢ (𝑥 = 𝑧 → (𝐻‘𝑥) = (𝐻‘𝑧))
13	11, 12	neeq12d 2992	. . 3 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) ≠ (𝐻‘𝑥) ↔ (𝐹‘𝑧) ≠ (𝐻‘𝑧)))
14	2, 3, 4, 10, 13	cbvrabw 3433	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐻‘𝑥)} = {𝑧 ∈ 𝐴 ∣ (𝐹‘𝑧) ≠ (𝐻‘𝑧)}
15	1, 14	eqtri 2758	1 ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ (𝐹‘𝑧) ≠ (𝐻‘𝑧)}

Syntax hints:   → wi 4  ∀wal 1540   = wceq 1542   ∈ wcel 2114   ≠ wne 2931  {crab 3398  ‘cfv 6491

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-iota 6447  df-fv 6499

This theorem is referenced by:  bnj1523  35206