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Theorem cbval 2418
Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2392. Check out cbvalw 2043, cbvalvw 2044, cbvalv1 2363 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 232 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
51, 2, 4cbv3 2417 . 2 (∀𝑥𝜑 → ∀𝑦𝜓)
63biimprd 251 . . . 4 (𝑥 = 𝑦 → (𝜓𝜑))
76equcoms 2028 . . 3 (𝑦 = 𝑥 → (𝜓𝜑))
82, 1, 7cbv3 2417 . 2 (∀𝑦𝜓 → ∀𝑥𝜑)
95, 8impbii 212 1 (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536  wnf 1785
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-11 2162  ax-12 2179  ax-13 2392
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786
This theorem is referenced by:  cbvex  2419  cbvalv  2420  cbval2  2434  cbval2OLD  2435  sb8  2561
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