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Theorem sbcop1 5344
Description: The proper substitution of an ordered pair for a setvar variable corresponds to a proper substitution of its first component. (Contributed by AV, 8-Apr-2023.)
Hypothesis
Ref Expression
sbcop.z (𝑧 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))
Assertion
Ref Expression
sbcop1 ([𝑎 / 𝑥]𝜓[𝑎, 𝑦⟩ / 𝑧]𝜑)
Distinct variable groups:   𝑥,𝑎,𝑦,𝑧   𝜑,𝑥,𝑦   𝜓,𝑧
Allowed substitution hints:   𝜑(𝑧,𝑎)   𝜓(𝑥,𝑦,𝑎)

Proof of Theorem sbcop1
StepHypRef Expression
1 sbc5 3748 . . . . 5 ([𝑎 / 𝑥]𝜓 ↔ ∃𝑥(𝑥 = 𝑎𝜓))
2 opeq1 4763 . . . . . . . . . . 11 (𝑎 = 𝑥 → ⟨𝑎, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
32equcoms 2027 . . . . . . . . . 10 (𝑥 = 𝑎 → ⟨𝑎, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
43eqeq2d 2809 . . . . . . . . 9 (𝑥 = 𝑎 → (𝑧 = ⟨𝑎, 𝑦⟩ ↔ 𝑧 = ⟨𝑥, 𝑦⟩))
5 sbcop.z . . . . . . . . . 10 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))
65biimprd 251 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝜓𝜑))
74, 6syl6bi 256 . . . . . . . 8 (𝑥 = 𝑎 → (𝑧 = ⟨𝑎, 𝑦⟩ → (𝜓𝜑)))
87com23 86 . . . . . . 7 (𝑥 = 𝑎 → (𝜓 → (𝑧 = ⟨𝑎, 𝑦⟩ → 𝜑)))
98imp 410 . . . . . 6 ((𝑥 = 𝑎𝜓) → (𝑧 = ⟨𝑎, 𝑦⟩ → 𝜑))
109exlimiv 1931 . . . . 5 (∃𝑥(𝑥 = 𝑎𝜓) → (𝑧 = ⟨𝑎, 𝑦⟩ → 𝜑))
111, 10sylbi 220 . . . 4 ([𝑎 / 𝑥]𝜓 → (𝑧 = ⟨𝑎, 𝑦⟩ → 𝜑))
1211alrimiv 1928 . . 3 ([𝑎 / 𝑥]𝜓 → ∀𝑧(𝑧 = ⟨𝑎, 𝑦⟩ → 𝜑))
13 opex 5321 . . . 4 𝑎, 𝑦⟩ ∈ V
1413sbc6 3750 . . 3 ([𝑎, 𝑦⟩ / 𝑧]𝜑 ↔ ∀𝑧(𝑧 = ⟨𝑎, 𝑦⟩ → 𝜑))
1512, 14sylibr 237 . 2 ([𝑎 / 𝑥]𝜓[𝑎, 𝑦⟩ / 𝑧]𝜑)
16 sbc5 3748 . . 3 ([𝑎, 𝑦⟩ / 𝑧]𝜑 ↔ ∃𝑧(𝑧 = ⟨𝑎, 𝑦⟩ ∧ 𝜑))
175biimpd 232 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))
184, 17syl6bi 256 . . . . . . . 8 (𝑥 = 𝑎 → (𝑧 = ⟨𝑎, 𝑦⟩ → (𝜑𝜓)))
1918com3l 89 . . . . . . 7 (𝑧 = ⟨𝑎, 𝑦⟩ → (𝜑 → (𝑥 = 𝑎𝜓)))
2019imp 410 . . . . . 6 ((𝑧 = ⟨𝑎, 𝑦⟩ ∧ 𝜑) → (𝑥 = 𝑎𝜓))
2120alrimiv 1928 . . . . 5 ((𝑧 = ⟨𝑎, 𝑦⟩ ∧ 𝜑) → ∀𝑥(𝑥 = 𝑎𝜓))
22 vex 3444 . . . . . 6 𝑎 ∈ V
2322sbc6 3750 . . . . 5 ([𝑎 / 𝑥]𝜓 ↔ ∀𝑥(𝑥 = 𝑎𝜓))
2421, 23sylibr 237 . . . 4 ((𝑧 = ⟨𝑎, 𝑦⟩ ∧ 𝜑) → [𝑎 / 𝑥]𝜓)
2524exlimiv 1931 . . 3 (∃𝑧(𝑧 = ⟨𝑎, 𝑦⟩ ∧ 𝜑) → [𝑎 / 𝑥]𝜓)
2616, 25sylbi 220 . 2 ([𝑎, 𝑦⟩ / 𝑧]𝜑[𝑎 / 𝑥]𝜓)
2715, 26impbii 212 1 ([𝑎 / 𝑥]𝜓[𝑎, 𝑦⟩ / 𝑧]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536   = wceq 1538  wex 1781  [wsbc 3720  cop 4531
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  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532
This theorem is referenced by:  sbcop  5345
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