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Theorem 2ralsng 4411
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 3167 . . 3 (𝑥 = 𝐴 → (∀𝑦 ∈ {𝐵}𝜑 ↔ ∀𝑦 ∈ {𝐵}𝜓))
32ralsng 4409 . 2 (𝐴𝑉 → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝜑 ↔ ∀𝑦 ∈ {𝐵}𝜓))
4 2ralsng.1 . . 3 (𝑦 = 𝐵 → (𝜓𝜒))
54ralsng 4409 . 2 (𝐵𝑊 → (∀𝑦 ∈ {𝐵}𝜓𝜒))
63, 5sylan9bb 506 1 ((𝐴𝑉𝐵𝑊) → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝜑𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wral 3089  {csn 4368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-v 3387  df-sbc 3634  df-sn 4369
This theorem is referenced by:  mat1ghm  20615  mat1mhm  20616  c0snmgmhm  42713  zrrnghm  42716
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