Theorem rexxpf 5745

Description: Version of rexxp 5740 with bound-variable hypotheses. (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
ralxpf.1	⊢ Ⅎ𝑦𝜑
ralxpf.2	⊢ Ⅎ𝑧𝜑
ralxpf.3	⊢ Ⅎ𝑥𝜓
ralxpf.4	⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
rexxpf	⊢ (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝜓)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑧,𝐵,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)   𝐴(𝑧)

Proof of Theorem rexxpf

Step	Hyp	Ref	Expression
1		ralxpf.1	. . . . . 6 ⊢ Ⅎ𝑦𝜑
2	1	nfn 1861	. . . . 5 ⊢ Ⅎ𝑦 ¬ 𝜑
3		ralxpf.2	. . . . . 6 ⊢ Ⅎ𝑧𝜑
4	3	nfn 1861	. . . . 5 ⊢ Ⅎ𝑧 ¬ 𝜑
5		ralxpf.3	. . . . . 6 ⊢ Ⅎ𝑥𝜓
6	5	nfn 1861	. . . . 5 ⊢ Ⅎ𝑥 ¬ 𝜓
7		ralxpf.4	. . . . . 6 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))
8	7	notbid 317	. . . . 5 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (¬ 𝜑 ↔ ¬ 𝜓))
9	2, 4, 6, 8	ralxpf 5744	. . . 4 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵) ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ¬ 𝜓)
10		ralnex 3163	. . . . 5 ⊢ (∀𝑧 ∈ 𝐵 ¬ 𝜓 ↔ ¬ ∃𝑧 ∈ 𝐵 𝜓)
11	10	ralbii 3090	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ¬ 𝜓 ↔ ∀𝑦 ∈ 𝐴 ¬ ∃𝑧 ∈ 𝐵 𝜓)
12	9, 11	bitri 274	. . 3 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵) ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐴 ¬ ∃𝑧 ∈ 𝐵 𝜓)
13	12	notbii 319	. 2 ⊢ (¬ ∀𝑥 ∈ (𝐴 × 𝐵) ¬ 𝜑 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ ∃𝑧 ∈ 𝐵 𝜓)
14		dfrex2 3166	. 2 ⊢ (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ¬ ∀𝑥 ∈ (𝐴 × 𝐵) ¬ 𝜑)
15		dfrex2 3166	. 2 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝜓 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ ∃𝑧 ∈ 𝐵 𝜓)
16	13, 14, 15	3bitr4i 302	1 ⊢ (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   = wceq 1539  Ⅎwnf 1787  ∀wral 3063  ∃wrex 3064  ⟨cop 4564   × cxp 5578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-iun 4923  df-opab 5133  df-xp 5586  df-rel 5587

This theorem is referenced by:  iunxpf  5746  wdom2d2  40773