Theorem mobiOLDOLD 2562

Description: Obsolete proof of mobi 2560 as of 15-Oct-2022. (Contributed by BJ, 7-Oct-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
mobiOLDOLD	⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃*𝑥𝜑 ↔ ∃*𝑥𝜓))

Proof of Theorem mobiOLDOLD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		imbi1 339	. . . . . 6 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 → 𝑥 = 𝑦) ↔ (𝜓 → 𝑥 = 𝑦)))
2	1	alimi 1855	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ∀𝑥((𝜑 → 𝑥 = 𝑦) ↔ (𝜓 → 𝑥 = 𝑦)))
3		albi 1862	. . . . 5 ⊢ (∀𝑥((𝜑 → 𝑥 = 𝑦) ↔ (𝜓 → 𝑥 = 𝑦)) → (∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥(𝜓 → 𝑥 = 𝑦)))
4	2, 3	syl 17	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥(𝜓 → 𝑥 = 𝑦)))
5	4	alrimiv 1970	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ∀𝑦(∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥(𝜓 → 𝑥 = 𝑦)))
6		exbi 1891	. . 3 ⊢ (∀𝑦(∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥(𝜓 → 𝑥 = 𝑦)) → (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∃𝑦∀𝑥(𝜓 → 𝑥 = 𝑦)))
7	5, 6	syl 17	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∃𝑦∀𝑥(𝜓 → 𝑥 = 𝑦)))
8		df-mo 2551	. 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
9		df-mo 2551	. 2 ⊢ (∃*𝑥𝜓 ↔ ∃𝑦∀𝑥(𝜓 → 𝑥 = 𝑦))
10	7, 8, 9	3bitr4g 306	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃*𝑥𝜑 ↔ ∃*𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1599  ∃wex 1823  ∃*wmo 2549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953

This theorem depends on definitions:  df-bi 199  df-ex 1824  df-mo 2551

This theorem is referenced by: (None)