Theorem mobidvALT 35041

Description: Alternate proof of mobidv 2549 directly from its analogues albidv 1923 and exbidv 1924, using deduction style. Note the proof structure, similar to mobi 2547. (Contributed by Mario Carneiro, 7-Oct-2016.) Reduce axiom dependencies and shorten proof. Remove dependency on ax-6 1971, ax-7 2011, ax-12 2171 by adapting proof of mobid 2550. (Revised by BJ, 26-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
mobidvALT.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))

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.

Step	Hyp	Ref	Expression
1		mobidvALT.1	. . . . 5 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	imbi1d 342	. . . 4 ⊢ (𝜑 → ((𝜓 → 𝑥 = 𝑦) ↔ (𝜒 → 𝑥 = 𝑦)))
3	2	albidv 1923	. . 3 ⊢ (𝜑 → (∀𝑥(𝜓 → 𝑥 = 𝑦) ↔ ∀𝑥(𝜒 → 𝑥 = 𝑦)))
4	3	exbidv 1924	. 2 ⊢ (𝜑 → (∃𝑦∀𝑥(𝜓 → 𝑥 = 𝑦) ↔ ∃𝑦∀𝑥(𝜒 → 𝑥 = 𝑦)))
5		df-mo 2540	. 2 ⊢ (∃*𝑥𝜓 ↔ ∃𝑦∀𝑥(𝜓 → 𝑥 = 𝑦))
6		df-mo 2540	. 2 ⊢ (∃*𝑥𝜒 ↔ ∃𝑦∀𝑥(𝜒 → 𝑥 = 𝑦))
7	4, 5, 6	3bitr4g 314	1 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537  ∃wex 1782  ∃*wmo 2538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913

This theorem depends on definitions:  df-bi 206  df-ex 1783  df-mo 2540

This theorem is referenced by: (None)