Theorem mobidvALT 37377

Description: Alternate proof of mobidv 2583 directly from its analogues albidv 1947 and exbidv 1948, using deduction style. Note the proof structure, similar to mobi 2581. (Contributed by Mario Carneiro, 7-Oct-2016.) Reduce axiom dependencies and shorten proof. Remove dependency on ax-12 2219 by adapting proof of mobid 2584. (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 344	. . . 4 ⊢ (𝜑 → ((𝜓 → 𝑥 = 𝑦) ↔ (𝜒 → 𝑥 = 𝑦)))
3	2	albidv 1947	. . 3 ⊢ (𝜑 → (∀𝑥(𝜓 → 𝑥 = 𝑦) ↔ ∀𝑥(𝜒 → 𝑥 = 𝑦)))
4	3	exbidv 1948	. 2 ⊢ (𝜑 → (∃𝑦∀𝑥(𝜓 → 𝑥 = 𝑦) ↔ ∃𝑦∀𝑥(𝜒 → 𝑥 = 𝑦)))
5		dfmo 2574	. 2 ⊢ (∃*𝑥𝜓 ↔ ∃𝑦∀𝑥(𝜓 → 𝑥 = 𝑦))
6		dfmo 2574	. 2 ⊢ (∃*𝑥𝜒 ↔ ∃𝑦∀𝑥(𝜒 → 𝑥 = 𝑦))
7	4, 5, 6	3bitr4g 317	1 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1565  ∃wex 1806  ∃*wmo 2571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-mo 2573

This theorem is referenced by: (None)