Theorem mo4OLD 2627

Description: Obsolete version of mo4 2625 as of 18-Oct-2023. (Contributed by NM, 26-Jul-1995.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
mo4OLD.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
mo4OLD	⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦   𝜓,𝑥

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

Proof of Theorem mo4OLD

Step	Hyp	Ref	Expression
1		nfv 1915	. 2 ⊢ Ⅎ𝑥𝜓
2		mo4OLD.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	1, 2	mo4f 2626	1 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃*wmo 2596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-11 2158  ax-12 2175

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598

This theorem is referenced by: (None)