Theorem 2sbc5g 45359

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 2773	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑤 = 𝑦 ↔ 𝑤 = 𝐵))
2	1	anbi2d 642	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ↔ (𝑧 = 𝑥 ∧ 𝑤 = 𝐵)))
3	2	anbi1d 643	. . . . 5 ⊢ (𝑦 = 𝐵 → (((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) ↔ ((𝑧 = 𝑥 ∧ 𝑤 = 𝐵) ∧ 𝜑)))
4	3	2exbidv 1957	. . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑧∃𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) ↔ ∃𝑧∃𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝐵) ∧ 𝜑)))
5		dfsbcq 3741	. . . . 5 ⊢ (𝑦 = 𝐵 → ([𝑦 / 𝑤]𝜑 ↔ [𝐵 / 𝑤]𝜑))
6	5	sbcbidv 3794	. . . 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 2773	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑧 = 𝑥 ↔ 𝑧 = 𝐴))
9	8	anbi1d 643	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑧 = 𝑥 ∧ 𝑤 = 𝐵) ↔ (𝑧 = 𝐴 ∧ 𝑤 = 𝐵)))
10	9	anbi1d 643	. . . . 5 ⊢ (𝑥 = 𝐴 → (((𝑧 = 𝑥 ∧ 𝑤 = 𝐵) ∧ 𝜑) ↔ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) ∧ 𝜑)))
11	10	2exbidv 1957	. . . 4 ⊢ (𝑥 = 𝐴 → (∃𝑧∃𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝐵) ∧ 𝜑) ↔ ∃𝑧∃𝑤((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) ∧ 𝜑)))
12		dfsbcq 3741	. . . 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		sbc5 3767	. . . 4 ⊢ ([𝑥 / 𝑧][𝑦 / 𝑤]𝜑 ↔ ∃𝑧(𝑧 = 𝑥 ∧ [𝑦 / 𝑤]𝜑))
15		19.42v 1986	. . . . . 6 ⊢ (∃𝑤(𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ 𝜑)) ↔ (𝑧 = 𝑥 ∧ ∃𝑤(𝑤 = 𝑦 ∧ 𝜑)))
16		anass 474	. . . . . . 7 ⊢ (((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) ↔ (𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ 𝜑)))
17	16	exbii 1881	. . . . . 6 ⊢ (∃𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) ↔ ∃𝑤(𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ 𝜑)))
18		sbc5 3767	. . . . . . 7 ⊢ ([𝑦 / 𝑤]𝜑 ↔ ∃𝑤(𝑤 = 𝑦 ∧ 𝜑))
19	18	anbi2i 635	. . . . . 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 1881	. . . 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 3534	. 2 ⊢ ((𝐵 ∈ 𝐷 ∧ 𝐴 ∈ 𝐶) → (∃𝑧∃𝑤((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) ∧ 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))
24	23	ancoms 464	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∃𝑧∃𝑤((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) ∧ 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  [wsbc 3739

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-sbc 3740

This theorem is used by:  pm14.123b  45369