Theorem 2sbc6g 41922

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 2750	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑤 = 𝑦 ↔ 𝑤 = 𝐵))
2	1	anbi2d 628	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ↔ (𝑧 = 𝑥 ∧ 𝑤 = 𝐵)))
3	2	imbi1d 341	. . . . 5 ⊢ (𝑦 = 𝐵 → (((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑) ↔ ((𝑧 = 𝑥 ∧ 𝑤 = 𝐵) → 𝜑)))
4	3	2albidv 1927	. . . 4 ⊢ (𝑦 = 𝐵 → (∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑) ↔ ∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝐵) → 𝜑)))
5		dfsbcq 3713	. . . . 5 ⊢ (𝑦 = 𝐵 → ([𝑦 / 𝑤]𝜑 ↔ [𝐵 / 𝑤]𝜑))
6	5	sbcbidv 3770	. . . 4 ⊢ (𝑦 = 𝐵 → ([𝑥 / 𝑧][𝑦 / 𝑤]𝜑 ↔ [𝑥 / 𝑧][𝐵 / 𝑤]𝜑))
7	4, 6	bibi12d 345	. . 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 2750	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑧 = 𝑥 ↔ 𝑧 = 𝐴))
9	8	anbi1d 629	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑧 = 𝑥 ∧ 𝑤 = 𝐵) ↔ (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)))
10	9	imbi1d 341	. . . . 5 ⊢ (𝑥 = 𝐴 → (((𝑧 = 𝑥 ∧ 𝑤 = 𝐵) → 𝜑) ↔ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → 𝜑)))
11	10	2albidv 1927	. . . 4 ⊢ (𝑥 = 𝐴 → (∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝐵) → 𝜑) ↔ ∀𝑧∀𝑤((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → 𝜑)))
12		dfsbcq 3713	. . . 4 ⊢ (𝑥 = 𝐴 → ([𝑥 / 𝑧][𝐵 / 𝑤]𝜑 ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))
13	11, 12	bibi12d 345	. . 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 3426	. . . . 5 ⊢ 𝑥 ∈ V
15	14	sbc6 3743	. . . 4 ⊢ ([𝑥 / 𝑧][𝑦 / 𝑤]𝜑 ↔ ∀𝑧(𝑧 = 𝑥 → [𝑦 / 𝑤]𝜑))
16		19.21v 1943	. . . . . 6 ⊢ (∀𝑤(𝑧 = 𝑥 → (𝑤 = 𝑦 → 𝜑)) ↔ (𝑧 = 𝑥 → ∀𝑤(𝑤 = 𝑦 → 𝜑)))
17		impexp 450	. . . . . . 7 ⊢ (((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑) ↔ (𝑧 = 𝑥 → (𝑤 = 𝑦 → 𝜑)))
18	17	albii 1823	. . . . . 6 ⊢ (∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑) ↔ ∀𝑤(𝑧 = 𝑥 → (𝑤 = 𝑦 → 𝜑)))
19		vex 3426	. . . . . . . 8 ⊢ 𝑦 ∈ V
20	19	sbc6 3743	. . . . . . 7 ⊢ ([𝑦 / 𝑤]𝜑 ↔ ∀𝑤(𝑤 = 𝑦 → 𝜑))
21	20	imbi2i 335	. . . . . 6 ⊢ ((𝑧 = 𝑥 → [𝑦 / 𝑤]𝜑) ↔ (𝑧 = 𝑥 → ∀𝑤(𝑤 = 𝑦 → 𝜑)))
22	16, 18, 21	3bitr4ri 303	. . . . 5 ⊢ ((𝑧 = 𝑥 → [𝑦 / 𝑤]𝜑) ↔ ∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑))
23	22	albii 1823	. . . 4 ⊢ (∀𝑧(𝑧 = 𝑥 → [𝑦 / 𝑤]𝜑) ↔ ∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑))
24	15, 23	bitr2i 275	. . 3 ⊢ (∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑) ↔ [𝑥 / 𝑧][𝑦 / 𝑤]𝜑)
25	7, 13, 24	vtocl2g 3500	. 2 ⊢ ((𝐵 ∈ 𝐷 ∧ 𝐴 ∈ 𝐶) → (∀𝑧∀𝑤((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))
26	25	ancoms 458	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∀𝑧∀𝑤((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537   = wceq 1539   ∈ wcel 2108  [wsbc 3711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-sbc 3712

This theorem is referenced by:  pm14.123a  41932