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Theorem cbvaldva 2384
Description: Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.) Remove dependency on ax-10 2183. (Revised by Wolf Lammen, 18-Jul-2021.)
Hypothesis
Ref Expression
cbvaldva.1 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
Assertion
Ref Expression
cbvaldva (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Distinct variable groups:   𝜓,𝑦   𝜒,𝑥   𝜑,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)

Proof of Theorem cbvaldva
StepHypRef Expression
1 cbvaldva.1 . . . . . 6 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
21expcom 402 . . . . 5 (𝑥 = 𝑦 → (𝜑 → (𝜓𝜒)))
32pm5.74d 264 . . . 4 (𝑥 = 𝑦 → ((𝜑𝜓) ↔ (𝜑𝜒)))
43cbvalv 2378 . . 3 (∀𝑥(𝜑𝜓) ↔ ∀𝑦(𝜑𝜒))
5 19.21v 2034 . . 3 (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))
6 19.21v 2034 . . 3 (∀𝑦(𝜑𝜒) ↔ (𝜑 → ∀𝑦𝜒))
74, 5, 63bitr3i 292 . 2 ((𝜑 → ∀𝑥𝜓) ↔ (𝜑 → ∀𝑦𝜒))
87pm5.74ri 263 1 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-11 2198  ax-12 2211  ax-13 2352
This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1875  df-nf 1879
This theorem is referenced by:  cbvexdva  2385  cbval2v  2386  cbvraldva2  3323
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