Theorem rexxpf 5861

Description: Version of rexxp 5856 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 1855	. . . . 5 ⊢ Ⅎ𝑦 ¬ 𝜑
3		ralxpf.2	. . . . . 6 ⊢ Ⅎ𝑧𝜑
4	3	nfn 1855	. . . . 5 ⊢ Ⅎ𝑧 ¬ 𝜑
5		ralxpf.3	. . . . . 6 ⊢ Ⅎ𝑥𝜓
6	5	nfn 1855	. . . . 5 ⊢ Ⅎ𝑥 ¬ 𝜓
7		ralxpf.4	. . . . . 6 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))
8	7	notbid 318	. . . . 5 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (¬ 𝜑 ↔ ¬ 𝜓))
9	2, 4, 6, 8	ralxpf 5860	. . . 4 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵) ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ¬ 𝜓)
10		ralnex 3070	. . . . 5 ⊢ (∀𝑧 ∈ 𝐵 ¬ 𝜓 ↔ ¬ ∃𝑧 ∈ 𝐵 𝜓)
11	10	ralbii 3091	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ¬ 𝜓 ↔ ∀𝑦 ∈ 𝐴 ¬ ∃𝑧 ∈ 𝐵 𝜓)
12	9, 11	bitri 275	. . 3 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵) ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐴 ¬ ∃𝑧 ∈ 𝐵 𝜓)
13	12	notbii 320	. 2 ⊢ (¬ ∀𝑥 ∈ (𝐴 × 𝐵) ¬ 𝜑 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ ∃𝑧 ∈ 𝐵 𝜓)
14		dfrex2 3071	. 2 ⊢ (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ¬ ∀𝑥 ∈ (𝐴 × 𝐵) ¬ 𝜑)
15		dfrex2 3071	. 2 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝜓 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ ∃𝑧 ∈ 𝐵 𝜓)
16	13, 14, 15	3bitr4i 303	1 ⊢ (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1537  Ⅎwnf 1780  ∀wral 3059  ∃wrex 3068  ⟨cop 4637   × cxp 5687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-iun 4998  df-opab 5211  df-xp 5695  df-rel 5696

This theorem is referenced by:  iunxpf  5862  wdom2d2  43024