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Theorem bj-dral1v 32873
Description: Version of dral1 2356 with a dv condition, which does not require ax-13 2282. Remark: the corresponding versions for dral2 2355 and drex2 2359 are instances of albidv 1889 and exbidv 1890 respectively. (Contributed by BJ, 17-Jun-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-dral1v.1 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
bj-dral1v (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem bj-dral1v
StepHypRef Expression
1 nfa1 2068 . . 3 𝑥𝑥 𝑥 = 𝑦
2 bj-dral1v.1 . . 3 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
31, 2albid 2128 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑥𝜓))
4 bj-axc11v 32872 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 → ∀𝑦𝜓))
5 axc11r 2223 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜓 → ∀𝑥𝜓))
64, 5impbid 202 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 ↔ ∀𝑦𝜓))
73, 6bitrd 268 1 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750
This theorem is referenced by:  bj-drex1v  32874  bj-drnf1v  32875
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