Theorem 2sbc6g 40740

Description: Theorem *13.21 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)

Assertion

Ref	Expression
2sbc6g	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∀𝑧∀𝑤((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))

Distinct variable groups:   𝑧,𝑤,𝐴   𝑤,𝐵,𝑧

Allowed substitution hints:   𝜑(𝑧,𝑤)   𝐶(𝑧,𝑤)   𝐷(𝑧,𝑤)

Proof of Theorem 2sbc6g

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq2 2833	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑤 = 𝑦 ↔ 𝑤 = 𝐵))
2	1	anbi2d 630	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ↔ (𝑧 = 𝑥 ∧ 𝑤 = 𝐵)))
3	2	imbi1d 344	. . . . 5 ⊢ (𝑦 = 𝐵 → (((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑) ↔ ((𝑧 = 𝑥 ∧ 𝑤 = 𝐵) → 𝜑)))
4	3	2albidv 1920	. . . 4 ⊢ (𝑦 = 𝐵 → (∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑) ↔ ∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝐵) → 𝜑)))
5		dfsbcq 3773	. . . . 5 ⊢ (𝑦 = 𝐵 → ([𝑦 / 𝑤]𝜑 ↔ [𝐵 / 𝑤]𝜑))
6	5	sbcbidv 3826	. . . 4 ⊢ (𝑦 = 𝐵 → ([𝑥 / 𝑧][𝑦 / 𝑤]𝜑 ↔ [𝑥 / 𝑧][𝐵 / 𝑤]𝜑))
7	4, 6	bibi12d 348	. . 3 ⊢ (𝑦 = 𝐵 → ((∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑) ↔ [𝑥 / 𝑧][𝑦 / 𝑤]𝜑) ↔ (∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝐵) → 𝜑) ↔ [𝑥 / 𝑧][𝐵 / 𝑤]𝜑)))
8		eqeq2 2833	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑧 = 𝑥 ↔ 𝑧 = 𝐴))
9	8	anbi1d 631	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑧 = 𝑥 ∧ 𝑤 = 𝐵) ↔ (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)))
10	9	imbi1d 344	. . . . 5 ⊢ (𝑥 = 𝐴 → (((𝑧 = 𝑥 ∧ 𝑤 = 𝐵) → 𝜑) ↔ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → 𝜑)))
11	10	2albidv 1920	. . . 4 ⊢ (𝑥 = 𝐴 → (∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝐵) → 𝜑) ↔ ∀𝑧∀𝑤((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → 𝜑)))
12		dfsbcq 3773	. . . 4 ⊢ (𝑥 = 𝐴 → ([𝑥 / 𝑧][𝐵 / 𝑤]𝜑 ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))
13	11, 12	bibi12d 348	. . 3 ⊢ (𝑥 = 𝐴 → ((∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝐵) → 𝜑) ↔ [𝑥 / 𝑧][𝐵 / 𝑤]𝜑) ↔ (∀𝑧∀𝑤((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑)))
14		vex 3497	. . . . 5 ⊢ 𝑥 ∈ V
15	14	sbc6 3801	. . . 4 ⊢ ([𝑥 / 𝑧][𝑦 / 𝑤]𝜑 ↔ ∀𝑧(𝑧 = 𝑥 → [𝑦 / 𝑤]𝜑))
16		19.21v 1936	. . . . . 6 ⊢ (∀𝑤(𝑧 = 𝑥 → (𝑤 = 𝑦 → 𝜑)) ↔ (𝑧 = 𝑥 → ∀𝑤(𝑤 = 𝑦 → 𝜑)))
17		impexp 453	. . . . . . 7 ⊢ (((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑) ↔ (𝑧 = 𝑥 → (𝑤 = 𝑦 → 𝜑)))
18	17	albii 1816	. . . . . 6 ⊢ (∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑) ↔ ∀𝑤(𝑧 = 𝑥 → (𝑤 = 𝑦 → 𝜑)))
19		vex 3497	. . . . . . . 8 ⊢ 𝑦 ∈ V
20	19	sbc6 3801	. . . . . . 7 ⊢ ([𝑦 / 𝑤]𝜑 ↔ ∀𝑤(𝑤 = 𝑦 → 𝜑))
21	20	imbi2i 338	. . . . . 6 ⊢ ((𝑧 = 𝑥 → [𝑦 / 𝑤]𝜑) ↔ (𝑧 = 𝑥 → ∀𝑤(𝑤 = 𝑦 → 𝜑)))
22	16, 18, 21	3bitr4ri 306	. . . . 5 ⊢ ((𝑧 = 𝑥 → [𝑦 / 𝑤]𝜑) ↔ ∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑))
23	22	albii 1816	. . . 4 ⊢ (∀𝑧(𝑧 = 𝑥 → [𝑦 / 𝑤]𝜑) ↔ ∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑))
24	15, 23	bitr2i 278	. . 3 ⊢ (∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑) ↔ [𝑥 / 𝑧][𝑦 / 𝑤]𝜑)
25	7, 13, 24	vtocl2g 3571	. 2 ⊢ ((𝐵 ∈ 𝐷 ∧ 𝐴 ∈ 𝐶) → (∀𝑧∀𝑤((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))
26	25	ancoms 461	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∀𝑧∀𝑤((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1531   = wceq 1533   ∈ wcel 2110  [wsbc 3771

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-v 3496  df-sbc 3772

This theorem is referenced by:  pm14.123a  40750