Theorem 2sbc6g 38616

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 2633	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑤 = 𝑦 ↔ 𝑤 = 𝐵))
2	1	anbi2d 740	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ↔ (𝑧 = 𝑥 ∧ 𝑤 = 𝐵)))
3	2	imbi1d 331	. . . . 5 ⊢ (𝑦 = 𝐵 → (((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑) ↔ ((𝑧 = 𝑥 ∧ 𝑤 = 𝐵) → 𝜑)))
4	3	2albidv 1851	. . . 4 ⊢ (𝑦 = 𝐵 → (∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑) ↔ ∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝐵) → 𝜑)))
5		dfsbcq 3437	. . . . 5 ⊢ (𝑦 = 𝐵 → ([𝑦 / 𝑤]𝜑 ↔ [𝐵 / 𝑤]𝜑))
6	5	sbcbidv 3490	. . . 4 ⊢ (𝑦 = 𝐵 → ([𝑥 / 𝑧][𝑦 / 𝑤]𝜑 ↔ [𝑥 / 𝑧][𝐵 / 𝑤]𝜑))
7	4, 6	bibi12d 335	. . 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 2633	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑧 = 𝑥 ↔ 𝑧 = 𝐴))
9	8	anbi1d 741	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑧 = 𝑥 ∧ 𝑤 = 𝐵) ↔ (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)))
10	9	imbi1d 331	. . . . 5 ⊢ (𝑥 = 𝐴 → (((𝑧 = 𝑥 ∧ 𝑤 = 𝐵) → 𝜑) ↔ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → 𝜑)))
11	10	2albidv 1851	. . . 4 ⊢ (𝑥 = 𝐴 → (∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝐵) → 𝜑) ↔ ∀𝑧∀𝑤((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → 𝜑)))
12		dfsbcq 3437	. . . 4 ⊢ (𝑥 = 𝐴 → ([𝑥 / 𝑧][𝐵 / 𝑤]𝜑 ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))
13	11, 12	bibi12d 335	. . 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 3203	. . . . 5 ⊢ 𝑥 ∈ V
15	14	sbc6 3462	. . . 4 ⊢ ([𝑥 / 𝑧][𝑦 / 𝑤]𝜑 ↔ ∀𝑧(𝑧 = 𝑥 → [𝑦 / 𝑤]𝜑))
16		19.21v 1868	. . . . . 6 ⊢ (∀𝑤(𝑧 = 𝑥 → (𝑤 = 𝑦 → 𝜑)) ↔ (𝑧 = 𝑥 → ∀𝑤(𝑤 = 𝑦 → 𝜑)))
17		impexp 462	. . . . . . 7 ⊢ (((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑) ↔ (𝑧 = 𝑥 → (𝑤 = 𝑦 → 𝜑)))
18	17	albii 1747	. . . . . 6 ⊢ (∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑) ↔ ∀𝑤(𝑧 = 𝑥 → (𝑤 = 𝑦 → 𝜑)))
19		vex 3203	. . . . . . . 8 ⊢ 𝑦 ∈ V
20	19	sbc6 3462	. . . . . . 7 ⊢ ([𝑦 / 𝑤]𝜑 ↔ ∀𝑤(𝑤 = 𝑦 → 𝜑))
21	20	imbi2i 326	. . . . . 6 ⊢ ((𝑧 = 𝑥 → [𝑦 / 𝑤]𝜑) ↔ (𝑧 = 𝑥 → ∀𝑤(𝑤 = 𝑦 → 𝜑)))
22	16, 18, 21	3bitr4ri 293	. . . . 5 ⊢ ((𝑧 = 𝑥 → [𝑦 / 𝑤]𝜑) ↔ ∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑))
23	22	albii 1747	. . . 4 ⊢ (∀𝑧(𝑧 = 𝑥 → [𝑦 / 𝑤]𝜑) ↔ ∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑))
24	15, 23	bitr2i 265	. . 3 ⊢ (∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → 𝜑) ↔ [𝑥 / 𝑧][𝑦 / 𝑤]𝜑)
25	7, 13, 24	vtocl2g 3270	. 2 ⊢ ((𝐵 ∈ 𝐷 ∧ 𝐴 ∈ 𝐶) → (∀𝑧∀𝑤((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))
26	25	ancoms 469	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∀𝑧∀𝑤((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481   = wceq 1483   ∈ wcel 1990  [wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436

This theorem is referenced by:  pm14.123a  38626