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Theorem cbvral 3431
 Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2392. Use the weaker cbvralw 3426 when possible. (Contributed by NM, 31-Jul-2003.) (New usage is discouraged.)
Hypotheses
Ref Expression
cbvral.1 𝑦𝜑
cbvral.2 𝑥𝜓
cbvral.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvral (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbvral
StepHypRef Expression
1 nfcv 2982 . 2 𝑥𝐴
2 nfcv 2982 . 2 𝑦𝐴
3 cbvral.1 . 2 𝑦𝜑
4 cbvral.2 . 2 𝑥𝜓
5 cbvral.3 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
61, 2, 3, 4, 5cbvralf 3424 1 (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  Ⅎwnf 1785  ∀wral 3133 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-10 2146  ax-11 2162  ax-12 2179  ax-13 2392 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clel 2896  df-nfc 2964  df-ral 3138 This theorem is referenced by:  cbvralv  3438  cbvralsv  3455  cbviing  4951  ralrnmpt  6851
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