Theorem ralxpxfr2d 3327

Description: Transfer a universal quantifier between one variable with pair-like semantics and two. (Contributed by Stefan O'Rear, 27-Feb-2015.)

Hypotheses

Ref	Expression
ralxpxfr2d.a	⊢ 𝐴 ∈ V
ralxpxfr2d.b	⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝑥 = 𝐴))
ralxpxfr2d.c	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
ralxpxfr2d	⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 𝜒))

Distinct variable groups:   𝜑,𝑥,𝑧   𝜑,𝑦,𝑥   𝜓,𝑦   𝜓,𝑧   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝜒,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦,𝑧)   𝐴(𝑦,𝑧)   𝐵(𝑥,𝑦,𝑧)   𝐶(𝑦,𝑧)   𝐷(𝑦,𝑧)

Proof of Theorem ralxpxfr2d

Step	Hyp	Ref	Expression
1		df-ral 2917	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜓))
2		ralxpxfr2d.b	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝑥 = 𝐴))
3	2	imbi1d 331	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 → 𝜓) ↔ (∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝑥 = 𝐴 → 𝜓)))
4	3	albidv 1849	. . . 4 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐵 → 𝜓) ↔ ∀𝑥(∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝑥 = 𝐴 → 𝜓)))
5	1, 4	syl5bb 272	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑥(∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝑥 = 𝐴 → 𝜓)))
6		ralcom4 3224	. . . 4 ⊢ (∀𝑦 ∈ 𝐶 ∀𝑥∀𝑧 ∈ 𝐷 (𝑥 = 𝐴 → 𝜓) ↔ ∀𝑥∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 (𝑥 = 𝐴 → 𝜓))
7		ralcom4 3224	. . . . 5 ⊢ (∀𝑧 ∈ 𝐷 ∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ ∀𝑥∀𝑧 ∈ 𝐷 (𝑥 = 𝐴 → 𝜓))
8	7	ralbii 2980	. . . 4 ⊢ (∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 ∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ ∀𝑦 ∈ 𝐶 ∀𝑥∀𝑧 ∈ 𝐷 (𝑥 = 𝐴 → 𝜓))
9		r19.23v 3023	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝐷 (𝑥 = 𝐴 → 𝜓) ↔ (∃𝑧 ∈ 𝐷 𝑥 = 𝐴 → 𝜓))
10	9	ralbii 2980	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 (𝑥 = 𝐴 → 𝜓) ↔ ∀𝑦 ∈ 𝐶 (∃𝑧 ∈ 𝐷 𝑥 = 𝐴 → 𝜓))
11		r19.23v 3023	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐶 (∃𝑧 ∈ 𝐷 𝑥 = 𝐴 → 𝜓) ↔ (∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝑥 = 𝐴 → 𝜓))
12	10, 11	bitr2i 265	. . . . 5 ⊢ ((∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝑥 = 𝐴 → 𝜓) ↔ ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 (𝑥 = 𝐴 → 𝜓))
13	12	albii 1747	. . . 4 ⊢ (∀𝑥(∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝑥 = 𝐴 → 𝜓) ↔ ∀𝑥∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 (𝑥 = 𝐴 → 𝜓))
14	6, 8, 13	3bitr4ri 293	. . 3 ⊢ (∀𝑥(∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝑥 = 𝐴 → 𝜓) ↔ ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 ∀𝑥(𝑥 = 𝐴 → 𝜓))
15	5, 14	syl6bb 276	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 ∀𝑥(𝑥 = 𝐴 → 𝜓)))
16		ralxpxfr2d.c	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
17	16	pm5.74da 723	. . . . 5 ⊢ (𝜑 → ((𝑥 = 𝐴 → 𝜓) ↔ (𝑥 = 𝐴 → 𝜒)))
18	17	albidv 1849	. . . 4 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ ∀𝑥(𝑥 = 𝐴 → 𝜒)))
19		ralxpxfr2d.a	. . . . 5 ⊢ 𝐴 ∈ V
20		biidd 252	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜒 ↔ 𝜒))
21	19, 20	ceqsalv 3233	. . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜒) ↔ 𝜒)
22	18, 21	syl6bb 276	. . 3 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ 𝜒))
23	22	2ralbidv 2989	. 2 ⊢ (𝜑 → (∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 ∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 𝜒))
24	15, 23	bitrd 268	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 𝜒))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202

This theorem is referenced by:  ralxpmap  7907