Theorem opelopabsb 5531

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 3479	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
2		vex 3479	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
3	1, 2	opnzi 5475	. . . . . . . . 9 ⊢ ⟨𝑥, 𝑦⟩ ≠ ∅
4		simpl 484	. . . . . . . . . . 11 ⊢ ((∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∅ = ⟨𝑥, 𝑦⟩)
5	4	eqcomd 2739	. . . . . . . . . 10 ⊢ ((∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ⟨𝑥, 𝑦⟩ = ∅)
6	5	necon3ai 2966	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ≠ ∅ → ¬ (∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
7	3, 6	ax-mp 5	. . . . . . . 8 ⊢ ¬ (∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
8	7	nex 1803	. . . . . . 7 ⊢ ¬ ∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
9	8	nex 1803	. . . . . 6 ⊢ ¬ ∃𝑥∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
10		elopab 5528	. . . . . 6 ⊢ (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
11	9, 10	mtbir 323	. . . . 5 ⊢ ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
12		eleq1 2822	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ = ∅ → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
13	11, 12	mtbiri 327	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ = ∅ → ¬ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
14	13	necon2ai 2971	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ⟨𝐴, 𝐵⟩ ≠ ∅)
15		opnz 5474	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ≠ ∅ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
16	14, 15	sylib 217	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (𝐴 ∈ V ∧ 𝐵 ∈ V))
17		sbcex 3788	. . 3 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 → 𝐴 ∈ V)
18		spesbc 3877	. . . 4 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 → ∃𝑥[𝐵 / 𝑦]𝜑)
19		sbcex 3788	. . . . 5 ⊢ ([𝐵 / 𝑦]𝜑 → 𝐵 ∈ V)
20	19	exlimiv 1934	. . . 4 ⊢ (∃𝑥[𝐵 / 𝑦]𝜑 → 𝐵 ∈ V)
21	18, 20	syl 17	. . 3 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 → 𝐵 ∈ V)
22	17, 21	jca 513	. 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
23		opeq1 4874	. . . . 5 ⊢ (𝑧 = 𝐴 → ⟨𝑧, 𝑤⟩ = ⟨𝐴, 𝑤⟩)
24	23	eleq1d 2819	. . . 4 ⊢ (𝑧 = 𝐴 → (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝐴, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
25		dfsbcq2 3781	. . . 4 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝐴 / 𝑥][𝑤 / 𝑦]𝜑))
26	24, 25	bibi12d 346	. . 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 4875	. . . . 5 ⊢ (𝑤 = 𝐵 → ⟨𝐴, 𝑤⟩ = ⟨𝐴, 𝐵⟩)
28	27	eleq1d 2819	. . . 4 ⊢ (𝑤 = 𝐵 → (⟨𝐴, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
29		dfsbcq2 3781	. . . . 5 ⊢ (𝑤 = 𝐵 → ([𝑤 / 𝑦]𝜑 ↔ [𝐵 / 𝑦]𝜑))
30	29	sbcbidv 3837	. . . 4 ⊢ (𝑤 = 𝐵 → ([𝐴 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑))
31	28, 30	bibi12d 346	. . 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 5530	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
33	26, 31, 32	vtocl2g 3563	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑))
34	16, 22, 33	pm5.21nii 380	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 397   = wceq 1542  ∃wex 1782  [wsb 2068   ∈ wcel 2107   ≠ wne 2941  Vcvv 3475  [wsbc 3778  ∅c0 4323  ⟨cop 4635  {copab 5211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-opab 5212

This theorem is referenced by:  brabsb  5532  opelopabgf  5541  opelopabaf  5545  opelopabf  5546  difopabOLD  5832  isarep1OLD  6639  fmptsnd  7167