Theorem mo4fOLD 2584

Description: Obsolete version of mo4f 2583 as of 19-Jan-2023. (Contributed by NM, 10-Apr-2004.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
mo4f.1	⊢ Ⅎ𝑥𝜓
mo4f.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
mo4fOLD	⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)

Proof of Theorem mo4fOLD

Step	Hyp	Ref	Expression
1		nfv 1957	. . 3 ⊢ Ⅎ𝑦𝜑
2	1	mo3OLD 2581	. 2 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
3		mo4f.1	. . . . . 6 ⊢ Ⅎ𝑥𝜓
4		mo4f.2	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	3, 4	sbie 2484	. . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
6	5	anbi2i 616	. . . 4 ⊢ ((𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝜑 ∧ 𝜓))
7	6	imbi1i 341	. . 3 ⊢ (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
8	7	2albii 1864	. 2 ⊢ (∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
9	2, 8	bitri 267	1 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386  ∀wal 1599  Ⅎwnf 1827  [wsb 2011  ∃*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  ax-6 2021  ax-7 2055  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551

This theorem is referenced by: (None)