Theorem opelopabsb 5508

Description: The law of concretion in terms of substitutions. (Contributed by NM, 30-Sep-2002.) (Revised by Mario Carneiro, 18-Nov-2016.)

Assertion

Ref	Expression
opelopabsb	⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)

Distinct variable groups:   𝑥,𝑦   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝐴(𝑥, 𝑦)   𝐵(𝑦)

Proof of Theorem opelopabsb

Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3458	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
2		vex 3458	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
3	1, 2	opnzi 5447	. . . . . . . . 9 ⊢ ⟨𝑥, 𝑦⟩ ≠ ∅
4		simpl 488	. . . . . . . . . . 11 ⊢ ((∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∅ = ⟨𝑥, 𝑦⟩)
5	4	eqcomd 2769	. . . . . . . . . 10 ⊢ ((∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ⟨𝑥, 𝑦⟩ = ∅)
6	5	necon3ai 2984	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ≠ ∅ → ¬ (∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
7	3, 6	ax-mp 5	. . . . . . . 8 ⊢ ¬ (∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
8	7	nex 1833	. . . . . . 7 ⊢ ¬ ∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
9	8	nex 1833	. . . . . 6 ⊢ ¬ ∃𝑥∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
10		elopab 5505	. . . . . 6 ⊢ (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
11	9, 10	mtbir 326	. . . . 5 ⊢ ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
12		eleq1 2852	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ = ∅ → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
13	11, 12	mtbiri 330	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ = ∅ → ¬ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
14	13	necon2ai 2988	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ⟨𝐴, 𝐵⟩ ≠ ∅)
15		opnz 5446	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ≠ ∅ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
16	14, 15	sylib 221	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (𝐴 ∈ V ∧ 𝐵 ∈ V))
17		sbcex 3752	. . 3 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 → 𝐴 ∈ V)
18		spesbc 3833	. . . 4 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 → ∃𝑥[𝐵 / 𝑦]𝜑)
19		sbcex 3752	. . . . 5 ⊢ ([𝐵 / 𝑦]𝜑 → 𝐵 ∈ V)
20	19	exlimiv 1965	. . . 4 ⊢ (∃𝑥[𝐵 / 𝑦]𝜑 → 𝐵 ∈ V)
21	18, 20	syl 18	. . 3 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 → 𝐵 ∈ V)
22	17, 21	jca 521	. 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
23		opeq1 4843	. . . . 5 ⊢ (𝑧 = 𝐴 → ⟨𝑧, 𝑤⟩ = ⟨𝐴, 𝑤⟩)
24	23	eleq1d 2849	. . . 4 ⊢ (𝑧 = 𝐴 → (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝐴, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
25		dfsbcq2 3745	. . . 4 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝐴 / 𝑥][𝑤 / 𝑦]𝜑))
26	24, 25	bibi12d 348	. . 3 ⊢ (𝑧 = 𝐴 → ((⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ (⟨𝐴, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝑤 / 𝑦]𝜑)))
27		opeq2 4844	. . . . 5 ⊢ (𝑤 = 𝐵 → ⟨𝐴, 𝑤⟩ = ⟨𝐴, 𝐵⟩)
28	27	eleq1d 2849	. . . 4 ⊢ (𝑤 = 𝐵 → (⟨𝐴, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
29		dfsbcq2 3745	. . . . 5 ⊢ (𝑤 = 𝐵 → ([𝑤 / 𝑦]𝜑 ↔ [𝐵 / 𝑦]𝜑))
30	29	sbcbidv 3797	. . . 4 ⊢ (𝑤 = 𝐵 → ([𝐴 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑))
31	28, 30	bibi12d 348	. . 3 ⊢ (𝑤 = 𝐵 → ((⟨𝐴, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝑤 / 𝑦]𝜑) ↔ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)))
32		vopelopabsb 5507	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
33	26, 31, 32	vtocl2g 3537	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑))
34	16, 22, 33	pm5.21nii 381	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812  [wsb 2101   ∈ wcel 2147   ≠ wne 2959  Vcvv 3454  [wsbc 3742  ∅c0 4281  ⟨cop 4838  {copab 5170

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-opab 5171

This theorem is used by:  brabsb  5509  opelopabgf  5519  opelopabaf  5523  opelopabf  5524  fmptsnd  7197