Theorem mobidvALT

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

	Ref	Expression
Hypothesis	mobidvALT.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion	mobidvALT	⊢ ( 𝜑 → ( ∃* 𝑥 𝜓 ↔ ∃* 𝑥 𝜒 ) )

Proof

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