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Theorem mobidvALT 34240
Description: Alternate proof of mobidv 2634 directly from its analogues albidv 1922 and exbidv 1923, using deduction style. Note the proof structure, similar to mobi 2631. (Contributed by Mario Carneiro, 7-Oct-2016.) Reduce axiom dependencies and shorten proof. Remove dependency on ax-6 1971, ax-7 2016, ax-12 2179 by adapting proof of mobid 2635. (Revised by BJ, 26-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
mobidvALT.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
mobidvALT (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem mobidvALT
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 mobidvALT.1 . . . . 5 (𝜑 → (𝜓𝜒))
21imbi1d 345 . . . 4 (𝜑 → ((𝜓𝑥 = 𝑦) ↔ (𝜒𝑥 = 𝑦)))
32albidv 1922 . . 3 (𝜑 → (∀𝑥(𝜓𝑥 = 𝑦) ↔ ∀𝑥(𝜒𝑥 = 𝑦)))
43exbidv 1923 . 2 (𝜑 → (∃𝑦𝑥(𝜓𝑥 = 𝑦) ↔ ∃𝑦𝑥(𝜒𝑥 = 𝑦)))
5 df-mo 2624 . 2 (∃*𝑥𝜓 ↔ ∃𝑦𝑥(𝜓𝑥 = 𝑦))
6 df-mo 2624 . 2 (∃*𝑥𝜒 ↔ ∃𝑦𝑥(𝜒𝑥 = 𝑦))
74, 5, 63bitr4g 317 1 (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536  wex 1781  ∃*wmo 2622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912
This theorem depends on definitions:  df-bi 210  df-ex 1782  df-mo 2624
This theorem is referenced by: (None)
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