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Theorem 2ralsng 4359
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 3135 . . 3 (𝑥 = 𝐴 → (∀𝑦 ∈ {𝐵}𝜑 ↔ ∀𝑦 ∈ {𝐵}𝜓))
32ralsng 4357 . 2 (𝐴𝑉 → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝜑 ↔ ∀𝑦 ∈ {𝐵}𝜓))
4 2ralsng.1 . . 3 (𝑦 = 𝐵 → (𝜓𝜒))
54ralsng 4357 . 2 (𝐵𝑊 → (∀𝑦 ∈ {𝐵}𝜓𝜒))
63, 5sylan9bb 495 1 ((𝐴𝑉𝐵𝑊) → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝜑𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  {csn 4317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-v 3353  df-sbc 3589  df-sn 4318
This theorem is referenced by:  mat1ghm  20508  mat1mhm  20509  c0snmgmhm  42443  zrrnghm  42446
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