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Theorem rexbidar 37671
 Description: More general form of rexbida 3029. (Contributed by Andrew Salmon, 25-Jul-2011.)
Hypotheses
Ref Expression
ralbidar.1 (𝜑 → ∀𝑥𝐴 𝜑)
ralbidar.2 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
rexbidar (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))

Proof of Theorem rexbidar
StepHypRef Expression
1 ralbidar.1 . . . . 5 (𝜑 → ∀𝑥𝐴 𝜑)
2 ralbidar.2 . . . . . . 7 ((𝜑𝑥𝐴) → (𝜓𝜒))
32ex 449 . . . . . 6 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
43ralimi 2936 . . . . 5 (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 (𝑥𝐴 → (𝜓𝜒)))
51, 4syl 17 . . . 4 (𝜑 → ∀𝑥𝐴 (𝑥𝐴 → (𝜓𝜒)))
6 df-ral 2901 . . . 4 (∀𝑥𝐴 (𝑥𝐴 → (𝜓𝜒)) ↔ ∀𝑥(𝑥𝐴 → (𝑥𝐴 → (𝜓𝜒))))
75, 6sylib 207 . . 3 (𝜑 → ∀𝑥(𝑥𝐴 → (𝑥𝐴 → (𝜓𝜒))))
8 pm2.43 54 . . . . 5 ((𝑥𝐴 → (𝑥𝐴 → (𝜓𝜒))) → (𝑥𝐴 → (𝜓𝜒)))
98pm5.32d 669 . . . 4 ((𝑥𝐴 → (𝑥𝐴 → (𝜓𝜒))) → ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
109alimi 1730 . . 3 (∀𝑥(𝑥𝐴 → (𝑥𝐴 → (𝜓𝜒))) → ∀𝑥((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
11 exbi 1762 . . 3 (∀𝑥((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)) → (∃𝑥(𝑥𝐴𝜓) ↔ ∃𝑥(𝑥𝐴𝜒)))
127, 10, 113syl 18 . 2 (𝜑 → (∃𝑥(𝑥𝐴𝜓) ↔ ∃𝑥(𝑥𝐴𝜒)))
13 df-rex 2902 . 2 (∃𝑥𝐴 𝜓 ↔ ∃𝑥(𝑥𝐴𝜓))
14 df-rex 2902 . 2 (∃𝑥𝐴 𝜒 ↔ ∃𝑥(𝑥𝐴𝜒))
1512, 13, 143bitr4g 302 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-ral 2901  df-rex 2902 This theorem is referenced by: (None)
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