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Theorem cbval 2393
Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2367. Check out cbvalw 2031, cbvalvw 2032, cbvalv1 2333 for versions requiring fewer axioms. (Contributed by NM, 13-May-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (New usage is discouraged.)
Hypotheses
Ref Expression
cbval.1 𝑦𝜑
cbval.2 𝑥𝜓
cbval.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbval (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Proof of Theorem cbval
StepHypRef Expression
1 cbval.1 . . 3 𝑦𝜑
2 cbval.2 . . 3 𝑥𝜓
3 cbval.3 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
43biimpd 228 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
51, 2, 4cbv3 2392 . 2 (∀𝑥𝜑 → ∀𝑦𝜓)
63biimprd 247 . . . 4 (𝑥 = 𝑦 → (𝜓𝜑))
76equcoms 2016 . . 3 (𝑦 = 𝑥 → (𝜓𝜑))
82, 1, 7cbv3 2392 . 2 (∀𝑦𝜓 → ∀𝑥𝜑)
95, 8impbii 208 1 (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1532  wnf 1778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-11 2147  ax-12 2167  ax-13 2367
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-nf 1779
This theorem is referenced by:  cbvex  2394  cbvalv  2395  cbval2  2406  sb8  2512
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