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Theorem opsbc2ie 30290
 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⟩ → (𝜑𝜒))
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
StepHypRef Expression
1 opsbc2ie.a . . . . . . . . 9 (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑𝜒))
21sbcth 3737 . . . . . . . 8 (𝑥 ∈ V → [𝑥 / 𝑎](𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑𝜒)))
3 sbcim1 3774 . . . . . . . 8 ([𝑥 / 𝑎](𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑𝜒)) → ([𝑥 / 𝑎]𝑝 = ⟨𝑎, 𝑏⟩ → [𝑥 / 𝑎](𝜑𝜒)))
42, 3syl 17 . . . . . . 7 (𝑥 ∈ V → ([𝑥 / 𝑎]𝑝 = ⟨𝑎, 𝑏⟩ → [𝑥 / 𝑎](𝜑𝜒)))
5 sbceq2g 4327 . . . . . . . 8 (𝑥 ∈ V → ([𝑥 / 𝑎]𝑝 = ⟨𝑎, 𝑏⟩ ↔ 𝑝 = 𝑥 / 𝑎𝑎, 𝑏⟩))
6 csbopg 4787 . . . . . . . . . 10 (𝑥 ∈ V → 𝑥 / 𝑎𝑎, 𝑏⟩ = ⟨𝑥 / 𝑎𝑎, 𝑥 / 𝑎𝑏⟩)
7 csbvarg 4342 . . . . . . . . . . 11 (𝑥 ∈ V → 𝑥 / 𝑎𝑎 = 𝑥)
8 csbconstg 3849 . . . . . . . . . . 11 (𝑥 ∈ V → 𝑥 / 𝑎𝑏 = 𝑏)
97, 8opeq12d 4777 . . . . . . . . . 10 (𝑥 ∈ V → ⟨𝑥 / 𝑎𝑎, 𝑥 / 𝑎𝑏⟩ = ⟨𝑥, 𝑏⟩)
106, 9eqtrd 2833 . . . . . . . . 9 (𝑥 ∈ V → 𝑥 / 𝑎𝑎, 𝑏⟩ = ⟨𝑥, 𝑏⟩)
1110eqeq2d 2809 . . . . . . . 8 (𝑥 ∈ V → (𝑝 = 𝑥 / 𝑎𝑎, 𝑏⟩ ↔ 𝑝 = ⟨𝑥, 𝑏⟩))
125, 11bitrd 282 . . . . . . 7 (𝑥 ∈ V → ([𝑥 / 𝑎]𝑝 = ⟨𝑎, 𝑏⟩ ↔ 𝑝 = ⟨𝑥, 𝑏⟩))
13 sbcbig 3772 . . . . . . . 8 (𝑥 ∈ V → ([𝑥 / 𝑎](𝜑𝜒) ↔ ([𝑥 / 𝑎]𝜑[𝑥 / 𝑎]𝜒)))
14 sbcg 3795 . . . . . . . . 9 (𝑥 ∈ V → ([𝑥 / 𝑎]𝜑𝜑))
1514bibi1d 347 . . . . . . . 8 (𝑥 ∈ V → (([𝑥 / 𝑎]𝜑[𝑥 / 𝑎]𝜒) ↔ (𝜑[𝑥 / 𝑎]𝜒)))
1613, 15bitrd 282 . . . . . . 7 (𝑥 ∈ V → ([𝑥 / 𝑎](𝜑𝜒) ↔ (𝜑[𝑥 / 𝑎]𝜒)))
174, 12, 163imtr3d 296 . . . . . 6 (𝑥 ∈ V → (𝑝 = ⟨𝑥, 𝑏⟩ → (𝜑[𝑥 / 𝑎]𝜒)))
1817elv 3447 . . . . 5 (𝑝 = ⟨𝑥, 𝑏⟩ → (𝜑[𝑥 / 𝑎]𝜒))
1918sbcth 3737 . . . 4 (𝑦 ∈ V → [𝑦 / 𝑏](𝑝 = ⟨𝑥, 𝑏⟩ → (𝜑[𝑥 / 𝑎]𝜒)))
20 sbcim1 3774 . . . 4 ([𝑦 / 𝑏](𝑝 = ⟨𝑥, 𝑏⟩ → (𝜑[𝑥 / 𝑎]𝜒)) → ([𝑦 / 𝑏]𝑝 = ⟨𝑥, 𝑏⟩ → [𝑦 / 𝑏](𝜑[𝑥 / 𝑎]𝜒)))
2119, 20syl 17 . . 3 (𝑦 ∈ V → ([𝑦 / 𝑏]𝑝 = ⟨𝑥, 𝑏⟩ → [𝑦 / 𝑏](𝜑[𝑥 / 𝑎]𝜒)))
22 sbceq2g 4327 . . . 4 (𝑦 ∈ V → ([𝑦 / 𝑏]𝑝 = ⟨𝑥, 𝑏⟩ ↔ 𝑝 = 𝑦 / 𝑏𝑥, 𝑏⟩))
23 csbopg 4787 . . . . . 6 (𝑦 ∈ V → 𝑦 / 𝑏𝑥, 𝑏⟩ = ⟨𝑦 / 𝑏𝑥, 𝑦 / 𝑏𝑏⟩)
24 csbconstg 3849 . . . . . . 7 (𝑦 ∈ V → 𝑦 / 𝑏𝑥 = 𝑥)
25 csbvarg 4342 . . . . . . 7 (𝑦 ∈ V → 𝑦 / 𝑏𝑏 = 𝑦)
2624, 25opeq12d 4777 . . . . . 6 (𝑦 ∈ V → ⟨𝑦 / 𝑏𝑥, 𝑦 / 𝑏𝑏⟩ = ⟨𝑥, 𝑦⟩)
2723, 26eqtrd 2833 . . . . 5 (𝑦 ∈ V → 𝑦 / 𝑏𝑥, 𝑏⟩ = ⟨𝑥, 𝑦⟩)
2827eqeq2d 2809 . . . 4 (𝑦 ∈ V → (𝑝 = 𝑦 / 𝑏𝑥, 𝑏⟩ ↔ 𝑝 = ⟨𝑥, 𝑦⟩))
2922, 28bitrd 282 . . 3 (𝑦 ∈ V → ([𝑦 / 𝑏]𝑝 = ⟨𝑥, 𝑏⟩ ↔ 𝑝 = ⟨𝑥, 𝑦⟩))
30 sbcbig 3772 . . . 4 (𝑦 ∈ V → ([𝑦 / 𝑏](𝜑[𝑥 / 𝑎]𝜒) ↔ ([𝑦 / 𝑏]𝜑[𝑦 / 𝑏][𝑥 / 𝑎]𝜒)))
31 sbcg 3795 . . . . 5 (𝑦 ∈ V → ([𝑦 / 𝑏]𝜑𝜑))
3231bibi1d 347 . . . 4 (𝑦 ∈ V → (([𝑦 / 𝑏]𝜑[𝑦 / 𝑏][𝑥 / 𝑎]𝜒) ↔ (𝜑[𝑦 / 𝑏][𝑥 / 𝑎]𝜒)))
3330, 32bitrd 282 . . 3 (𝑦 ∈ V → ([𝑦 / 𝑏](𝜑[𝑥 / 𝑎]𝜒) ↔ (𝜑[𝑦 / 𝑏][𝑥 / 𝑎]𝜒)))
3421, 29, 333imtr3d 296 . 2 (𝑦 ∈ V → (𝑝 = ⟨𝑥, 𝑦⟩ → (𝜑[𝑦 / 𝑏][𝑥 / 𝑎]𝜒)))
3534elv 3447 1 (𝑝 = ⟨𝑥, 𝑦⟩ → (𝜑[𝑦 / 𝑏][𝑥 / 𝑎]𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  Vcvv 3442  [wsbc 3722  ⦋csb 3830  ⟨cop 4534 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535 This theorem is referenced by:  opreu2reuALT  30291
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