Theorem speimfw

Description: Specialization, with additional weakening (compared to 19.2 ) to allow bundling of x and y . Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017) (Proof shortened by Wolf Lammen, 5-Dec-2017)

	Ref	Expression
Hypothesis	speimfw.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
Assertion	speimfw	⊢ ( ¬ ∀ 𝑥 ¬ 𝑥 = 𝑦 → ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		speimfw.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
2		df-ex	⊢ ( ∃ 𝑥 𝑥 = 𝑦 ↔ ¬ ∀ 𝑥 ¬ 𝑥 = 𝑦 )
3	2	biimpri	⊢ ( ¬ ∀ 𝑥 ¬ 𝑥 = 𝑦 → ∃ 𝑥 𝑥 = 𝑦 )
4	1	com12	⊢ ( 𝜑 → ( 𝑥 = 𝑦 → 𝜓 ) )
5	4	aleximi	⊢ ( ∀ 𝑥 𝜑 → ( ∃ 𝑥 𝑥 = 𝑦 → ∃ 𝑥 𝜓 ) )
6	3 5	syl5com	⊢ ( ¬ ∀ 𝑥 ¬ 𝑥 = 𝑦 → ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )