Metamath Proof Explorer


Theorem nfald2

Description: Variation on nfald which adds the hypothesis that x and y are distinct in the inner subproof. Usage of this theorem is discouraged because it depends on ax-13 . Check out nfald for a version requiring fewer axioms. (Contributed by Mario Carneiro, 8-Oct-2016) (New usage is discouraged.)

Ref Expression
Hypotheses nfald2.1 𝑦 𝜑
nfald2.2 ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑥 𝜓 )
Assertion nfald2 ( 𝜑 → Ⅎ 𝑥𝑦 𝜓 )

Proof

Step Hyp Ref Expression
1 nfald2.1 𝑦 𝜑
2 nfald2.2 ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑥 𝜓 )
3 nfnae 𝑦 ¬ ∀ 𝑥 𝑥 = 𝑦
4 1 3 nfan 𝑦 ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 )
5 4 2 nfald ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑥𝑦 𝜓 )
6 5 ex ( 𝜑 → ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥𝑦 𝜓 ) )
7 nfa1 𝑦𝑦 𝜓
8 biidd ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑦 𝜓 ↔ ∀ 𝑦 𝜓 ) )
9 8 drnf1 ( ∀ 𝑥 𝑥 = 𝑦 → ( Ⅎ 𝑥𝑦 𝜓 ↔ Ⅎ 𝑦𝑦 𝜓 ) )
10 7 9 mpbiri ( ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥𝑦 𝜓 )
11 6 10 pm2.61d2 ( 𝜑 → Ⅎ 𝑥𝑦 𝜓 )