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Theorem r2allem 3190
 Description: Lemma factoring out common proof steps of r2alf 3211 and r2al 3191. Introduced to reduce dependencies on axioms. (Contributed by Wolf Lammen, 9-Jan-2020.)
Hypothesis
Ref Expression
r2allem.1 (∀𝑦(𝑥𝐴 → (𝑦𝐵𝜑)) ↔ (𝑥𝐴 → ∀𝑦(𝑦𝐵𝜑)))
Assertion
Ref Expression
r2allem (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑))

Proof of Theorem r2allem
StepHypRef Expression
1 df-ral 3135 . 2 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐵 𝜑))
2 r2allem.1 . . . 4 (∀𝑦(𝑥𝐴 → (𝑦𝐵𝜑)) ↔ (𝑥𝐴 → ∀𝑦(𝑦𝐵𝜑)))
3 impexp 454 . . . . 5 (((𝑥𝐴𝑦𝐵) → 𝜑) ↔ (𝑥𝐴 → (𝑦𝐵𝜑)))
43albii 1821 . . . 4 (∀𝑦((𝑥𝐴𝑦𝐵) → 𝜑) ↔ ∀𝑦(𝑥𝐴 → (𝑦𝐵𝜑)))
5 df-ral 3135 . . . . 5 (∀𝑦𝐵 𝜑 ↔ ∀𝑦(𝑦𝐵𝜑))
65imbi2i 339 . . . 4 ((𝑥𝐴 → ∀𝑦𝐵 𝜑) ↔ (𝑥𝐴 → ∀𝑦(𝑦𝐵𝜑)))
72, 4, 63bitr4i 306 . . 3 (∀𝑦((𝑥𝐴𝑦𝐵) → 𝜑) ↔ (𝑥𝐴 → ∀𝑦𝐵 𝜑))
87albii 1821 . 2 (∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐵 𝜑))
91, 8bitr4i 281 1 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   ∈ wcel 2114  ∀wral 3130 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811 This theorem depends on definitions:  df-bi 210  df-an 400  df-ral 3135 This theorem is referenced by:  r2al  3191  r2alf  3211
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