Theorem 2sb5rf 1865

Description: Reversed double substitution. (Contributed by NM, 3-Feb-2005.)

Hypotheses

Ref	Expression
2sb5rf.1	⊢ (𝜑 → ∀𝑧𝜑)
2sb5rf.2	⊢ (𝜑 → ∀𝑤𝜑)

Assertion

Ref	Expression
2sb5rf	⊢ (𝜑 ↔ ∃𝑧∃𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))

Distinct variable groups:   𝑥,𝑦   𝑥,𝑤   𝑦,𝑧   𝑧,𝑤

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem 2sb5rf

Step	Hyp	Ref	Expression
1		2sb5rf.1	. . 3 ⊢ (𝜑 → ∀𝑧𝜑)
2	1	sb5rf 1732	. 2 ⊢ (𝜑 ↔ ∃𝑧(𝑧 = 𝑥 ∧ [𝑧 / 𝑥]𝜑))
3		19.42v 1786	. . . 4 ⊢ (∃𝑤(𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)) ↔ (𝑧 = 𝑥 ∧ ∃𝑤(𝑤 = 𝑦 ∧ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)))
4		sbcom2 1863	. . . . . . 7 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)
5	4	anbi2i 430	. . . . . 6 ⊢ (((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑))
6		anass 381	. . . . . 6 ⊢ (((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑) ↔ (𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)))
7	5, 6	bitri 173	. . . . 5 ⊢ (((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ (𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)))
8	7	exbii 1496	. . . 4 ⊢ (∃𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ ∃𝑤(𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)))
9		2sb5rf.2	. . . . . . 7 ⊢ (𝜑 → ∀𝑤𝜑)
10	9	hbsbv 1817	. . . . . 6 ⊢ ([𝑧 / 𝑥]𝜑 → ∀𝑤[𝑧 / 𝑥]𝜑)
11	10	sb5rf 1732	. . . . 5 ⊢ ([𝑧 / 𝑥]𝜑 ↔ ∃𝑤(𝑤 = 𝑦 ∧ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑))
12	11	anbi2i 430	. . . 4 ⊢ ((𝑧 = 𝑥 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑧 = 𝑥 ∧ ∃𝑤(𝑤 = 𝑦 ∧ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)))
13	3, 8, 12	3bitr4ri 202	. . 3 ⊢ ((𝑧 = 𝑥 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
14	13	exbii 1496	. 2 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃𝑧∃𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
15	2, 14	bitri 173	1 ⊢ (𝜑 ↔ ∃𝑧∃𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241  ∃wex 1381  [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646

This theorem is referenced by: (None)