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Theorem 2ralsng 4608
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 3195 . . 3 (𝑥 = 𝐴 → (∀𝑦 ∈ {𝐵}𝜑 ↔ ∀𝑦 ∈ {𝐵}𝜓))
32ralsng 4605 . 2 (𝐴𝑉 → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝜑 ↔ ∀𝑦 ∈ {𝐵}𝜓))
4 2ralsng.1 . . 3 (𝑦 = 𝐵 → (𝜓𝜒))
54ralsng 4605 . 2 (𝐵𝑊 → (∀𝑦 ∈ {𝐵}𝜓𝜒))
63, 5sylan9bb 512 1 ((𝐴𝑉𝐵𝑊) → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝜑𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1530  wcel 2107  wral 3136  {csn 4559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-v 3495  df-sbc 3771  df-sn 4560
This theorem is referenced by:  mat1ghm  21084  mat1mhm  21085  f1resfz0f1d  32354  c0snmgmhm  44175  zrrnghm  44178
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