Theorem 2sbc5g 41120

Description: Theorem *13.22 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)

Assertion

Ref	Expression
2sbc5g	⊢ ((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 2sbc5g

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

Step	Hyp	Ref	Expression
1		eqeq2 2810	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑤 = 𝑦 ↔ 𝑤 = 𝐵))
2	1	anbi2d 631	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ↔ (𝑧 = 𝑥 ∧ 𝑤 = 𝐵)))
3	2	anbi1d 632	. . . . 5 ⊢ (𝑦 = 𝐵 → (((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) ↔ ((𝑧 = 𝑥 ∧ 𝑤 = 𝐵) ∧ 𝜑)))
4	3	2exbidv 1925	. . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑧∃𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) ↔ ∃𝑧∃𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝐵) ∧ 𝜑)))
5		dfsbcq 3722	. . . . 5 ⊢ (𝑦 = 𝐵 → ([𝑦 / 𝑤]𝜑 ↔ [𝐵 / 𝑤]𝜑))
6	5	sbcbidv 3774	. . . 4 ⊢ (𝑦 = 𝐵 → ([𝑥 / 𝑧][𝑦 / 𝑤]𝜑 ↔ [𝑥 / 𝑧][𝐵 / 𝑤]𝜑))
7	4, 6	bibi12d 349	. . 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 2810	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑧 = 𝑥 ↔ 𝑧 = 𝐴))
9	8	anbi1d 632	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑧 = 𝑥 ∧ 𝑤 = 𝐵) ↔ (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)))
10	9	anbi1d 632	. . . . 5 ⊢ (𝑥 = 𝐴 → (((𝑧 = 𝑥 ∧ 𝑤 = 𝐵) ∧ 𝜑) ↔ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) ∧ 𝜑)))
11	10	2exbidv 1925	. . . 4 ⊢ (𝑥 = 𝐴 → (∃𝑧∃𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝐵) ∧ 𝜑) ↔ ∃𝑧∃𝑤((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) ∧ 𝜑)))
12		dfsbcq 3722	. . . 4 ⊢ (𝑥 = 𝐴 → ([𝑥 / 𝑧][𝐵 / 𝑤]𝜑 ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))
13	11, 12	bibi12d 349	. . 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		sbc5 3748	. . . 4 ⊢ ([𝑥 / 𝑧][𝑦 / 𝑤]𝜑 ↔ ∃𝑧(𝑧 = 𝑥 ∧ [𝑦 / 𝑤]𝜑))
15		19.42v 1954	. . . . . 6 ⊢ (∃𝑤(𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ 𝜑)) ↔ (𝑧 = 𝑥 ∧ ∃𝑤(𝑤 = 𝑦 ∧ 𝜑)))
16		anass 472	. . . . . . 7 ⊢ (((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) ↔ (𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ 𝜑)))
17	16	exbii 1849	. . . . . 6 ⊢ (∃𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) ↔ ∃𝑤(𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ 𝜑)))
18		sbc5 3748	. . . . . . 7 ⊢ ([𝑦 / 𝑤]𝜑 ↔ ∃𝑤(𝑤 = 𝑦 ∧ 𝜑))
19	18	anbi2i 625	. . . . . 6 ⊢ ((𝑧 = 𝑥 ∧ [𝑦 / 𝑤]𝜑) ↔ (𝑧 = 𝑥 ∧ ∃𝑤(𝑤 = 𝑦 ∧ 𝜑)))
20	15, 17, 19	3bitr4ri 307	. . . . 5 ⊢ ((𝑧 = 𝑥 ∧ [𝑦 / 𝑤]𝜑) ↔ ∃𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑))
21	20	exbii 1849	. . . 4 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ [𝑦 / 𝑤]𝜑) ↔ ∃𝑧∃𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑))
22	14, 21	bitr2i 279	. . 3 ⊢ (∃𝑧∃𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) ↔ [𝑥 / 𝑧][𝑦 / 𝑤]𝜑)
23	7, 13, 22	vtocl2g 3520	. 2 ⊢ ((𝐵 ∈ 𝐷 ∧ 𝐴 ∈ 𝐶) → (∃𝑧∃𝑤((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) ∧ 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))
24	23	ancoms 462	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∃𝑧∃𝑤((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) ∧ 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  [wsbc 3720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-sbc 3721

This theorem is referenced by:  pm14.123b  41130