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Theorem cbvalv1 2163
 Description: Version of cbval 2259 with a dv condition, which does not require ax-13 2234. See cbvalvw 1956 for a version with two dv conditions, requiring fewer axioms, and cbvalv 2261 for another variant. (Contributed by BJ, 31-May-2019.)
Hypotheses
Ref Expression
cbvalv1.nf1 𝑦𝜑
cbvalv1.nf2 𝑥𝜓
cbvalv1.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvalv1 (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbvalv1
StepHypRef Expression
1 cbvalv1.nf1 . . 3 𝑦𝜑
2 cbvalv1.nf2 . . 3 𝑥𝜓
3 cbvalv1.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
43biimpd 218 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
51, 2, 4cbv3v 2158 . 2 (∀𝑥𝜑 → ∀𝑦𝜓)
63biimprd 237 . . . 4 (𝑥 = 𝑦 → (𝜓𝜑))
76equcoms 1934 . . 3 (𝑦 = 𝑥 → (𝜓𝜑))
82, 1, 7cbv3v 2158 . 2 (∀𝑦𝜓 → ∀𝑥𝜑)
95, 8impbii 198 1 (∀𝑥𝜑 ↔ ∀𝑦𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  Ⅎwnf 1699 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-nf 1701 This theorem is referenced by:  cbvexv1  2164  cleqh  2711  bj-cbvalvv  31920  bj-cbval2v  31924  bj-abbi  31963
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