Theorem 2sb6rf 2478

Description: Reversed double substitution. Usage of this theorem is discouraged because it depends on ax-13 2377. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) Remove variable constraints. (Revised by Wolf Lammen, 28-Sep-2018.) (Proof shortened by Wolf Lammen, 13-Apr-2023.) (New usage is discouraged.)

Hypotheses

Ref	Expression
2sb5rf.1	⊢ Ⅎ𝑧𝜑
2sb5rf.2	⊢ Ⅎ𝑤𝜑

Assertion

Ref	Expression
2sb6rf	⊢ (𝜑 ↔ ∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))

Distinct variable group:   𝑧,𝑤

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

Proof of Theorem 2sb6rf

Step	Hyp	Ref	Expression
1		2sb5rf.1	. . . 4 ⊢ Ⅎ𝑧𝜑
2	1	19.23 2211	. . 3 ⊢ (∀𝑧(∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑) ↔ (∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑))
3		2sb5rf.2	. . . . 5 ⊢ Ⅎ𝑤𝜑
4	3	19.23 2211	. . . 4 ⊢ (∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑) ↔ (∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑))
5	4	albii 1819	. . 3 ⊢ (∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑) ↔ ∀𝑧(∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑))
6		2ax6e 2476	. . . 4 ⊢ ∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦)
7	6	a1bi 362	. . 3 ⊢ (𝜑 ↔ (∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑))
8	2, 5, 7	3bitr4ri 304	. 2 ⊢ (𝜑 ↔ ∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑))
9		sbequ12r 2252	. . . . 5 ⊢ (𝑧 = 𝑥 → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝑤 / 𝑦]𝜑))
10		sbequ12r 2252	. . . . 5 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑦]𝜑 ↔ 𝜑))
11	9, 10	sylan9bb 509	. . . 4 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ 𝜑))
12	11	pm5.74i 271	. . 3 ⊢ (((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑))
13	12	2albii 1820	. 2 ⊢ (∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ ∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑))
14	8, 13	bitr4i 278	1 ⊢ (𝜑 ↔ ∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538  ∃wex 1779  Ⅎwnf 1783  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2157  ax-12 2177  ax-13 2377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065

This theorem is referenced by: (None)