Theorem opsbc2ie 32151

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 3784	. . . . . . . 8 ⊢ (𝑥 ∈ V → [𝑥 / 𝑎](𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑 ↔ 𝜒)))
3		sbcim1 3825	. . . . . . . 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 4408	. . . . . . . 8 ⊢ (𝑥 ∈ V → ([𝑥 / 𝑎]𝑝 = ⟨𝑎, 𝑏⟩ ↔ 𝑝 = ⦋𝑥 / 𝑎⦌⟨𝑎, 𝑏⟩))
6		csbopg 4883	. . . . . . . . . 10 ⊢ (𝑥 ∈ V → ⦋𝑥 / 𝑎⦌⟨𝑎, 𝑏⟩ = ⟨⦋𝑥 / 𝑎⦌𝑎, ⦋𝑥 / 𝑎⦌𝑏⟩)
7		csbvarg 4423	. . . . . . . . . . 11 ⊢ (𝑥 ∈ V → ⦋𝑥 / 𝑎⦌𝑎 = 𝑥)
8		csbconstg 3904	. . . . . . . . . . 11 ⊢ (𝑥 ∈ V → ⦋𝑥 / 𝑎⦌𝑏 = 𝑏)
9	7, 8	opeq12d 4873	. . . . . . . . . 10 ⊢ (𝑥 ∈ V → ⟨⦋𝑥 / 𝑎⦌𝑎, ⦋𝑥 / 𝑎⦌𝑏⟩ = ⟨𝑥, 𝑏⟩)
10	6, 9	eqtrd 2764	. . . . . . . . 9 ⊢ (𝑥 ∈ V → ⦋𝑥 / 𝑎⦌⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑏⟩)
11	10	eqeq2d 2735	. . . . . . . 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 3823	. . . . . . . 8 ⊢ (𝑥 ∈ V → ([𝑥 / 𝑎](𝜑 ↔ 𝜒) ↔ ([𝑥 / 𝑎]𝜑 ↔ [𝑥 / 𝑎]𝜒)))
14		sbcg 3848	. . . . . . . . 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 3472	. . . . 5 ⊢ (𝑝 = ⟨𝑥, 𝑏⟩ → (𝜑 ↔ [𝑥 / 𝑎]𝜒))
19	18	sbcth 3784	. . . 4 ⊢ (𝑦 ∈ V → [𝑦 / 𝑏](𝑝 = ⟨𝑥, 𝑏⟩ → (𝜑 ↔ [𝑥 / 𝑎]𝜒)))
20		sbcim1 3825	. . . 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 4408	. . . 4 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑏]𝑝 = ⟨𝑥, 𝑏⟩ ↔ 𝑝 = ⦋𝑦 / 𝑏⦌⟨𝑥, 𝑏⟩))
23		csbopg 4883	. . . . . 6 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑏⦌⟨𝑥, 𝑏⟩ = ⟨⦋𝑦 / 𝑏⦌𝑥, ⦋𝑦 / 𝑏⦌𝑏⟩)
24		csbconstg 3904	. . . . . . 7 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑏⦌𝑥 = 𝑥)
25		csbvarg 4423	. . . . . . 7 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑏⦌𝑏 = 𝑦)
26	24, 25	opeq12d 4873	. . . . . 6 ⊢ (𝑦 ∈ V → ⟨⦋𝑦 / 𝑏⦌𝑥, ⦋𝑦 / 𝑏⦌𝑏⟩ = ⟨𝑥, 𝑦⟩)
27	23, 26	eqtrd 2764	. . . . 5 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑏⦌⟨𝑥, 𝑏⟩ = ⟨𝑥, 𝑦⟩)
28	27	eqeq2d 2735	. . . 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 3823	. . . 4 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑏](𝜑 ↔ [𝑥 / 𝑎]𝜒) ↔ ([𝑦 / 𝑏]𝜑 ↔ [𝑦 / 𝑏][𝑥 / 𝑎]𝜒)))
31		sbcg 3848	. . . . 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 3472	1 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ [𝑦 / 𝑏][𝑥 / 𝑎]𝜒))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098  Vcvv 3466  [wsbc 3769  ⦋csb 3885  ⟨cop 4626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627

This theorem is referenced by:  opreu2reuALT  32152