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Theorem 2ralsng 4683
Description: Substitution expressed in terms of two quantifications over singletons. (Contributed by AV, 22-Dec-2019.)
Hypotheses
Ref Expression
ralsng.1 (𝑥 = 𝐴 → (𝜑𝜓))
2ralsng.1 (𝑦 = 𝐵 → (𝜓𝜒))
Assertion
Ref Expression
2ralsng ((𝐴𝑉𝐵𝑊) → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝜑𝜒))
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥   𝑦,𝐴   𝑥,𝐵,𝑦   𝜒,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)   𝜒(𝑥)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem 2ralsng
StepHypRef Expression
1 ralsng.1 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
21ralbidv 3176 . . 3 (𝑥 = 𝐴 → (∀𝑦 ∈ {𝐵}𝜑 ↔ ∀𝑦 ∈ {𝐵}𝜓))
32ralsng 4680 . 2 (𝐴𝑉 → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝜑 ↔ ∀𝑦 ∈ {𝐵}𝜓))
4 2ralsng.1 . . 3 (𝑦 = 𝐵 → (𝜓𝜒))
54ralsng 4680 . 2 (𝐵𝑊 → (∀𝑦 ∈ {𝐵}𝜓𝜒))
63, 5sylan9bb 509 1 ((𝐴𝑉𝐵𝑊) → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝜑𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  {csn 4631
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-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-v 3480  df-sn 4632
This theorem is referenced by:  c0snmgmhm  20479  zrrnghm  20553  mat1ghm  22505  mat1mhm  22506  f1resfz0f1d  35098
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