Theorem 2sb6rf 2505

Description: Reversed double substitution. Usage of this theorem is discouraged because it depends on ax-13 2404. (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 2247	. . 3 ⊢ (∀𝑧(∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑) ↔ (∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑))
3		2sb5rf.2	. . . . 5 ⊢ Ⅎ𝑤𝜑
4	3	19.23 2247	. . . 4 ⊢ (∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑) ↔ (∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑))
5	4	albii 1849	. . 3 ⊢ (∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑) ↔ ∀𝑧(∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑))
6		2ax6e 2503	. . . 4 ⊢ ∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦)
7	6	a1bi 365	. . 3 ⊢ (𝜑 ↔ (∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑))
8	2, 5, 7	3bitr4ri 307	. 2 ⊢ (𝜑 ↔ ∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑))
9		sbequ12r 2288	. . . . 5 ⊢ (𝑧 = 𝑥 → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝑤 / 𝑦]𝜑))
10		sbequ12r 2288	. . . . 5 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑦]𝜑 ↔ 𝜑))
11	9, 10	sylan9bb 518	. . . 4 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ 𝜑))
12	11	pm5.74i 274	. . 3 ⊢ (((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑))
13	12	2albii 1850	. 2 ⊢ (∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ ∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑))
14	8, 13	bitr4i 281	1 ⊢ (𝜑 ↔ ∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1568  ∃wex 1809  Ⅎwnf 1813  [wsb 2096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-10 2176  ax-11 2192  ax-12 2213  ax-13 2404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097

This theorem is referenced by: (None)