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Theorem sbc2iedf 32494
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by Thierry Arnoux, 4-Jul-2023.)
Hypotheses
Ref Expression
sbc2iedf.1 𝑥𝜑
sbc2iedf.2 𝑦𝜑
sbc2iedf.3 𝑥𝜒
sbc2iedf.4 𝑦𝜒
sbc2iedf.5 (𝜑𝐴𝑉)
sbc2iedf.6 (𝜑𝐵𝑊)
sbc2iedf.7 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))
Assertion
Ref Expression
sbc2iedf (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓𝜒))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem sbc2iedf
StepHypRef Expression
1 sbc2iedf.5 . 2 (𝜑𝐴𝑉)
2 sbc2iedf.6 . . . 4 (𝜑𝐵𝑊)
32adantr 480 . . 3 ((𝜑𝑥 = 𝐴) → 𝐵𝑊)
4 sbc2iedf.7 . . . 4 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))
54anassrs 467 . . 3 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝜓𝜒))
6 sbc2iedf.2 . . . 4 𝑦𝜑
7 nfv 1912 . . . 4 𝑦 𝑥 = 𝐴
86, 7nfan 1897 . . 3 𝑦(𝜑𝑥 = 𝐴)
9 sbc2iedf.4 . . . 4 𝑦𝜒
109a1i 11 . . 3 ((𝜑𝑥 = 𝐴) → Ⅎ𝑦𝜒)
113, 5, 8, 10sbciedf 3836 . 2 ((𝜑𝑥 = 𝐴) → ([𝐵 / 𝑦]𝜓𝜒))
12 sbc2iedf.1 . 2 𝑥𝜑
13 sbc2iedf.3 . . 3 𝑥𝜒
1413a1i 11 . 2 (𝜑 → Ⅎ𝑥𝜒)
151, 11, 12, 14sbciedf 3836 1 (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wnf 1780  wcel 2106  [wsbc 3791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-sbc 3792
This theorem is referenced by:  rspc2daf  32495
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