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Theorem ralbidar 40645
Description: More general form of ralbida 3235. (Contributed by Andrew Salmon, 25-Jul-2011.)
Hypotheses
Ref Expression
ralbidar.1 (𝜑 → ∀𝑥𝐴 𝜑)
ralbidar.2 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
ralbidar (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 𝜒))

Proof of Theorem ralbidar
StepHypRef Expression
1 ralbidar.1 . . . . 5 (𝜑 → ∀𝑥𝐴 𝜑)
2 ralbidar.2 . . . . . . 7 ((𝜑𝑥𝐴) → (𝜓𝜒))
32ex 413 . . . . . 6 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
43ralimi 3165 . . . . 5 (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 (𝑥𝐴 → (𝜓𝜒)))
51, 4syl 17 . . . 4 (𝜑 → ∀𝑥𝐴 (𝑥𝐴 → (𝜓𝜒)))
6 df-ral 3148 . . . 4 (∀𝑥𝐴 (𝑥𝐴 → (𝜓𝜒)) ↔ ∀𝑥(𝑥𝐴 → (𝑥𝐴 → (𝜓𝜒))))
75, 6sylib 219 . . 3 (𝜑 → ∀𝑥(𝑥𝐴 → (𝑥𝐴 → (𝜓𝜒))))
8 pm2.43 56 . . . . 5 ((𝑥𝐴 → (𝑥𝐴 → (𝜓𝜒))) → (𝑥𝐴 → (𝜓𝜒)))
98pm5.74d 274 . . . 4 ((𝑥𝐴 → (𝑥𝐴 → (𝜓𝜒))) → ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
109alimi 1805 . . 3 (∀𝑥(𝑥𝐴 → (𝑥𝐴 → (𝜓𝜒))) → ∀𝑥((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
11 albi 1812 . . 3 (∀𝑥((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)) → (∀𝑥(𝑥𝐴𝜓) ↔ ∀𝑥(𝑥𝐴𝜒)))
127, 10, 113syl 18 . 2 (𝜑 → (∀𝑥(𝑥𝐴𝜓) ↔ ∀𝑥(𝑥𝐴𝜒)))
13 df-ral 3148 . 2 (∀𝑥𝐴 𝜓 ↔ ∀𝑥(𝑥𝐴𝜓))
14 df-ral 3148 . 2 (∀𝑥𝐴 𝜒 ↔ ∀𝑥(𝑥𝐴𝜒))
1512, 13, 143bitr4g 315 1 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1528  wcel 2107  wral 3143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803
This theorem depends on definitions:  df-bi 208  df-an 397  df-ral 3148
This theorem is referenced by: (None)
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