Theorem opsbc2ie 32504

Description: Conversion of implicit substitution to explicit class substitution for ordered pairs. (Contributed by Thierry Arnoux, 4-Jul-2023.)

Hypothesis

Ref	Expression
opsbc2ie.a	⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑 ↔ 𝜒))

Assertion

Ref	Expression
opsbc2ie	⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ [𝑦 / 𝑏][𝑥 / 𝑎]𝜒))

Distinct variable groups:   𝑎,𝑏,𝑝   𝜑,𝑎,𝑏   𝑥,𝑏

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑝)   𝜒(𝑥,𝑦,𝑝,𝑎,𝑏)

Proof of Theorem opsbc2ie

Step	Hyp	Ref	Expression
1		opsbc2ie.a	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑 ↔ 𝜒))
2	1	sbcth 3819	. . . . . . . 8 ⊢ (𝑥 ∈ V → [𝑥 / 𝑎](𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑 ↔ 𝜒)))
3		sbcim1 3861	. . . . . . . 8 ⊢ ([𝑥 / 𝑎](𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑 ↔ 𝜒)) → ([𝑥 / 𝑎]𝑝 = ⟨𝑎, 𝑏⟩ → [𝑥 / 𝑎](𝜑 ↔ 𝜒)))
4	2, 3	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ V → ([𝑥 / 𝑎]𝑝 = ⟨𝑎, 𝑏⟩ → [𝑥 / 𝑎](𝜑 ↔ 𝜒)))
5		sbceq2g 4442	. . . . . . . 8 ⊢ (𝑥 ∈ V → ([𝑥 / 𝑎]𝑝 = ⟨𝑎, 𝑏⟩ ↔ 𝑝 = ⦋𝑥 / 𝑎⦌⟨𝑎, 𝑏⟩))
6		csbopg 4915	. . . . . . . . . 10 ⊢ (𝑥 ∈ V → ⦋𝑥 / 𝑎⦌⟨𝑎, 𝑏⟩ = ⟨⦋𝑥 / 𝑎⦌𝑎, ⦋𝑥 / 𝑎⦌𝑏⟩)
7		csbvarg 4457	. . . . . . . . . . 11 ⊢ (𝑥 ∈ V → ⦋𝑥 / 𝑎⦌𝑎 = 𝑥)
8		csbconstg 3940	. . . . . . . . . . 11 ⊢ (𝑥 ∈ V → ⦋𝑥 / 𝑎⦌𝑏 = 𝑏)
9	7, 8	opeq12d 4905	. . . . . . . . . 10 ⊢ (𝑥 ∈ V → ⟨⦋𝑥 / 𝑎⦌𝑎, ⦋𝑥 / 𝑎⦌𝑏⟩ = ⟨𝑥, 𝑏⟩)
10	6, 9	eqtrd 2780	. . . . . . . . 9 ⊢ (𝑥 ∈ V → ⦋𝑥 / 𝑎⦌⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑏⟩)
11	10	eqeq2d 2751	. . . . . . . 8 ⊢ (𝑥 ∈ V → (𝑝 = ⦋𝑥 / 𝑎⦌⟨𝑎, 𝑏⟩ ↔ 𝑝 = ⟨𝑥, 𝑏⟩))
12	5, 11	bitrd 279	. . . . . . 7 ⊢ (𝑥 ∈ V → ([𝑥 / 𝑎]𝑝 = ⟨𝑎, 𝑏⟩ ↔ 𝑝 = ⟨𝑥, 𝑏⟩))
13		sbcbig 3859	. . . . . . . 8 ⊢ (𝑥 ∈ V → ([𝑥 / 𝑎](𝜑 ↔ 𝜒) ↔ ([𝑥 / 𝑎]𝜑 ↔ [𝑥 / 𝑎]𝜒)))
14		sbcg 3883	. . . . . . . . 9 ⊢ (𝑥 ∈ V → ([𝑥 / 𝑎]𝜑 ↔ 𝜑))
15	14	bibi1d 343	. . . . . . . 8 ⊢ (𝑥 ∈ V → (([𝑥 / 𝑎]𝜑 ↔ [𝑥 / 𝑎]𝜒) ↔ (𝜑 ↔ [𝑥 / 𝑎]𝜒)))
16	13, 15	bitrd 279	. . . . . . 7 ⊢ (𝑥 ∈ V → ([𝑥 / 𝑎](𝜑 ↔ 𝜒) ↔ (𝜑 ↔ [𝑥 / 𝑎]𝜒)))
17	4, 12, 16	3imtr3d 293	. . . . . 6 ⊢ (𝑥 ∈ V → (𝑝 = ⟨𝑥, 𝑏⟩ → (𝜑 ↔ [𝑥 / 𝑎]𝜒)))
18	17	elv 3493	. . . . 5 ⊢ (𝑝 = ⟨𝑥, 𝑏⟩ → (𝜑 ↔ [𝑥 / 𝑎]𝜒))
19	18	sbcth 3819	. . . 4 ⊢ (𝑦 ∈ V → [𝑦 / 𝑏](𝑝 = ⟨𝑥, 𝑏⟩ → (𝜑 ↔ [𝑥 / 𝑎]𝜒)))
20		sbcim1 3861	. . . 4 ⊢ ([𝑦 / 𝑏](𝑝 = ⟨𝑥, 𝑏⟩ → (𝜑 ↔ [𝑥 / 𝑎]𝜒)) → ([𝑦 / 𝑏]𝑝 = ⟨𝑥, 𝑏⟩ → [𝑦 / 𝑏](𝜑 ↔ [𝑥 / 𝑎]𝜒)))
21	19, 20	syl 17	. . 3 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑏]𝑝 = ⟨𝑥, 𝑏⟩ → [𝑦 / 𝑏](𝜑 ↔ [𝑥 / 𝑎]𝜒)))
22		sbceq2g 4442	. . . 4 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑏]𝑝 = ⟨𝑥, 𝑏⟩ ↔ 𝑝 = ⦋𝑦 / 𝑏⦌⟨𝑥, 𝑏⟩))
23		csbopg 4915	. . . . . 6 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑏⦌⟨𝑥, 𝑏⟩ = ⟨⦋𝑦 / 𝑏⦌𝑥, ⦋𝑦 / 𝑏⦌𝑏⟩)
24		csbconstg 3940	. . . . . . 7 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑏⦌𝑥 = 𝑥)
25		csbvarg 4457	. . . . . . 7 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑏⦌𝑏 = 𝑦)
26	24, 25	opeq12d 4905	. . . . . 6 ⊢ (𝑦 ∈ V → ⟨⦋𝑦 / 𝑏⦌𝑥, ⦋𝑦 / 𝑏⦌𝑏⟩ = ⟨𝑥, 𝑦⟩)
27	23, 26	eqtrd 2780	. . . . 5 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑏⦌⟨𝑥, 𝑏⟩ = ⟨𝑥, 𝑦⟩)
28	27	eqeq2d 2751	. . . 4 ⊢ (𝑦 ∈ V → (𝑝 = ⦋𝑦 / 𝑏⦌⟨𝑥, 𝑏⟩ ↔ 𝑝 = ⟨𝑥, 𝑦⟩))
29	22, 28	bitrd 279	. . 3 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑏]𝑝 = ⟨𝑥, 𝑏⟩ ↔ 𝑝 = ⟨𝑥, 𝑦⟩))
30		sbcbig 3859	. . . 4 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑏](𝜑 ↔ [𝑥 / 𝑎]𝜒) ↔ ([𝑦 / 𝑏]𝜑 ↔ [𝑦 / 𝑏][𝑥 / 𝑎]𝜒)))
31		sbcg 3883	. . . . 5 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑏]𝜑 ↔ 𝜑))
32	31	bibi1d 343	. . . 4 ⊢ (𝑦 ∈ V → (([𝑦 / 𝑏]𝜑 ↔ [𝑦 / 𝑏][𝑥 / 𝑎]𝜒) ↔ (𝜑 ↔ [𝑦 / 𝑏][𝑥 / 𝑎]𝜒)))
33	30, 32	bitrd 279	. . 3 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑏](𝜑 ↔ [𝑥 / 𝑎]𝜒) ↔ (𝜑 ↔ [𝑦 / 𝑏][𝑥 / 𝑎]𝜒)))
34	21, 29, 33	3imtr3d 293	. 2 ⊢ (𝑦 ∈ V → (𝑝 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ [𝑦 / 𝑏][𝑥 / 𝑎]𝜒)))
35	34	elv 3493	1 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ [𝑦 / 𝑏][𝑥 / 𝑎]𝜒))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2108  Vcvv 3488  [wsbc 3804  ⦋csb 3921  ⟨cop 4654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655

This theorem is referenced by:  opreu2reuALT  32505