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Theorem sbcpr 47445
Description: The proper substitution of an unordered pair for a setvar variable corresponds to a proper substitution of each of its elements. (Contributed by AV, 7-Apr-2023.)
Hypothesis
Ref Expression
sbcpr.x (𝑝 = {𝑥, 𝑦} → (𝜑𝜓))
Assertion
Ref Expression
sbcpr ([{𝑎, 𝑏} / 𝑝]𝜑[𝑏 / 𝑦][𝑎 / 𝑥]𝜓)
Distinct variable groups:   𝑎,𝑏,𝑥,𝑦,𝑝   𝜑,𝑥,𝑦   𝜓,𝑝
Allowed substitution hints:   𝜑(𝑝,𝑎,𝑏)   𝜓(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem sbcpr
StepHypRef Expression
1 sbc5 3818 . . 3 ([{𝑎, 𝑏} / 𝑝]𝜑 ↔ ∃𝑝(𝑝 = {𝑎, 𝑏} ∧ 𝜑))
2 preq12 4739 . . . . . . . . . . . . . . . . . 18 ((𝑥 = 𝑎𝑦 = 𝑏) → {𝑥, 𝑦} = {𝑎, 𝑏})
32eqcomd 2740 . . . . . . . . . . . . . . . . 17 ((𝑥 = 𝑎𝑦 = 𝑏) → {𝑎, 𝑏} = {𝑥, 𝑦})
43eqeq2d 2745 . . . . . . . . . . . . . . . 16 ((𝑥 = 𝑎𝑦 = 𝑏) → (𝑝 = {𝑎, 𝑏} ↔ 𝑝 = {𝑥, 𝑦}))
54biimpa 476 . . . . . . . . . . . . . . 15 (((𝑥 = 𝑎𝑦 = 𝑏) ∧ 𝑝 = {𝑎, 𝑏}) → 𝑝 = {𝑥, 𝑦})
6 sbcpr.x . . . . . . . . . . . . . . 15 (𝑝 = {𝑥, 𝑦} → (𝜑𝜓))
75, 6syl 17 . . . . . . . . . . . . . 14 (((𝑥 = 𝑎𝑦 = 𝑏) ∧ 𝑝 = {𝑎, 𝑏}) → (𝜑𝜓))
87biimpd 229 . . . . . . . . . . . . 13 (((𝑥 = 𝑎𝑦 = 𝑏) ∧ 𝑝 = {𝑎, 𝑏}) → (𝜑𝜓))
98expcom 413 . . . . . . . . . . . 12 (𝑝 = {𝑎, 𝑏} → ((𝑥 = 𝑎𝑦 = 𝑏) → (𝜑𝜓)))
109expd 415 . . . . . . . . . . 11 (𝑝 = {𝑎, 𝑏} → (𝑥 = 𝑎 → (𝑦 = 𝑏 → (𝜑𝜓))))
1110com24 95 . . . . . . . . . 10 (𝑝 = {𝑎, 𝑏} → (𝜑 → (𝑦 = 𝑏 → (𝑥 = 𝑎𝜓))))
1211imp31 417 . . . . . . . . 9 (((𝑝 = {𝑎, 𝑏} ∧ 𝜑) ∧ 𝑦 = 𝑏) → (𝑥 = 𝑎𝜓))
1312alrimiv 1924 . . . . . . . 8 (((𝑝 = {𝑎, 𝑏} ∧ 𝜑) ∧ 𝑦 = 𝑏) → ∀𝑥(𝑥 = 𝑎𝜓))
14 vex 3481 . . . . . . . . 9 𝑎 ∈ V
1514sbc6 3822 . . . . . . . 8 ([𝑎 / 𝑥]𝜓 ↔ ∀𝑥(𝑥 = 𝑎𝜓))
1613, 15sylibr 234 . . . . . . 7 (((𝑝 = {𝑎, 𝑏} ∧ 𝜑) ∧ 𝑦 = 𝑏) → [𝑎 / 𝑥]𝜓)
1716ex 412 . . . . . 6 ((𝑝 = {𝑎, 𝑏} ∧ 𝜑) → (𝑦 = 𝑏[𝑎 / 𝑥]𝜓))
1817alrimiv 1924 . . . . 5 ((𝑝 = {𝑎, 𝑏} ∧ 𝜑) → ∀𝑦(𝑦 = 𝑏[𝑎 / 𝑥]𝜓))
19 vex 3481 . . . . . 6 𝑏 ∈ V
2019sbc6 3822 . . . . 5 ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓 ↔ ∀𝑦(𝑦 = 𝑏[𝑎 / 𝑥]𝜓))
2118, 20sylibr 234 . . . 4 ((𝑝 = {𝑎, 𝑏} ∧ 𝜑) → [𝑏 / 𝑦][𝑎 / 𝑥]𝜓)
2221exlimiv 1927 . . 3 (∃𝑝(𝑝 = {𝑎, 𝑏} ∧ 𝜑) → [𝑏 / 𝑦][𝑎 / 𝑥]𝜓)
231, 22sylbi 217 . 2 ([{𝑎, 𝑏} / 𝑝]𝜑[𝑏 / 𝑦][𝑎 / 𝑥]𝜓)
24 sbc5 3818 . . . . 5 ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓 ↔ ∃𝑦(𝑦 = 𝑏[𝑎 / 𝑥]𝜓))
25 sbc5 3818 . . . . . . . 8 ([𝑎 / 𝑥]𝜓 ↔ ∃𝑥(𝑥 = 𝑎𝜓))
266bicomd 223 . . . . . . . . . . . . . . 15 (𝑝 = {𝑥, 𝑦} → (𝜓𝜑))
275, 26syl 17 . . . . . . . . . . . . . 14 (((𝑥 = 𝑎𝑦 = 𝑏) ∧ 𝑝 = {𝑎, 𝑏}) → (𝜓𝜑))
2827biimpd 229 . . . . . . . . . . . . 13 (((𝑥 = 𝑎𝑦 = 𝑏) ∧ 𝑝 = {𝑎, 𝑏}) → (𝜓𝜑))
2928expcom 413 . . . . . . . . . . . 12 (𝑝 = {𝑎, 𝑏} → ((𝑥 = 𝑎𝑦 = 𝑏) → (𝜓𝜑)))
3029com13 88 . . . . . . . . . . 11 (𝜓 → ((𝑥 = 𝑎𝑦 = 𝑏) → (𝑝 = {𝑎, 𝑏} → 𝜑)))
3130expd 415 . . . . . . . . . 10 (𝜓 → (𝑥 = 𝑎 → (𝑦 = 𝑏 → (𝑝 = {𝑎, 𝑏} → 𝜑))))
3231impcom 407 . . . . . . . . 9 ((𝑥 = 𝑎𝜓) → (𝑦 = 𝑏 → (𝑝 = {𝑎, 𝑏} → 𝜑)))
3332exlimiv 1927 . . . . . . . 8 (∃𝑥(𝑥 = 𝑎𝜓) → (𝑦 = 𝑏 → (𝑝 = {𝑎, 𝑏} → 𝜑)))
3425, 33sylbi 217 . . . . . . 7 ([𝑎 / 𝑥]𝜓 → (𝑦 = 𝑏 → (𝑝 = {𝑎, 𝑏} → 𝜑)))
3534impcom 407 . . . . . 6 ((𝑦 = 𝑏[𝑎 / 𝑥]𝜓) → (𝑝 = {𝑎, 𝑏} → 𝜑))
3635exlimiv 1927 . . . . 5 (∃𝑦(𝑦 = 𝑏[𝑎 / 𝑥]𝜓) → (𝑝 = {𝑎, 𝑏} → 𝜑))
3724, 36sylbi 217 . . . 4 ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓 → (𝑝 = {𝑎, 𝑏} → 𝜑))
3837alrimiv 1924 . . 3 ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓 → ∀𝑝(𝑝 = {𝑎, 𝑏} → 𝜑))
39 prex 5442 . . . 4 {𝑎, 𝑏} ∈ V
4039sbc6 3822 . . 3 ([{𝑎, 𝑏} / 𝑝]𝜑 ↔ ∀𝑝(𝑝 = {𝑎, 𝑏} → 𝜑))
4138, 40sylibr 234 . 2 ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓[{𝑎, 𝑏} / 𝑝]𝜑)
4223, 41impbii 209 1 ([{𝑎, 𝑏} / 𝑝]𝜑[𝑏 / 𝑦][𝑎 / 𝑥]𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1534   = wceq 1536  wex 1775  [wsbc 3790  {cpr 4632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-nul 4339  df-sn 4631  df-pr 4633
This theorem is referenced by:  reupr  47446
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