Theorem nfexd2

Description: Variation on nfexd 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 nfexd for a version requiring fewer axioms. (Contributed by Mario Carneiro, 8-Oct-2016) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nfald2.1	⊢ Ⅎ 𝑦 𝜑
	nfald2.2	⊢ ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑥 𝜓 )
Assertion	nfexd2	⊢ ( 𝜑 → Ⅎ 𝑥 ∃ 𝑦 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		nfald2.1	⊢ Ⅎ 𝑦 𝜑
2		nfald2.2	⊢ ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑥 𝜓 )
3		df-ex	⊢ ( ∃ 𝑦 𝜓 ↔ ¬ ∀ 𝑦 ¬ 𝜓 )
4	2	nfnd	⊢ ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑥 ¬ 𝜓 )
5	1 4	nfald2	⊢ ( 𝜑 → Ⅎ 𝑥 ∀ 𝑦 ¬ 𝜓 )
6	5	nfnd	⊢ ( 𝜑 → Ⅎ 𝑥 ¬ ∀ 𝑦 ¬ 𝜓 )
7	3 6	nfxfrd	⊢ ( 𝜑 → Ⅎ 𝑥 ∃ 𝑦 𝜓 )