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Metamath Proof Explorer

Theorem bj-exlimd

Description: A slightly more general exlimd . A common usage will have ph substituted for ps and th substituted for ta , giving a form closer to exlimd . (Contributed by BJ, 25-Dec-2023)

	Ref	Expression
Hypotheses	bj-exlimd.ph	⊢ ( 𝜑 → ∀ 𝑥 𝜓 )
	bj-exlimd.th	⊢ ( 𝜑 → ( ∃ 𝑥 𝜃 → 𝜏 ) )
	bj-exlimd.maj	⊢ ( 𝜓 → ( 𝜒 → 𝜃 ) )
Assertion	bj-exlimd	⊢ ( 𝜑 → ( ∃ 𝑥 𝜒 → 𝜏 ) )

Proof

Step	Hyp	Ref	Expression
1		bj-exlimd.ph	⊢ ( 𝜑 → ∀ 𝑥 𝜓 )
2		bj-exlimd.th	⊢ ( 𝜑 → ( ∃ 𝑥 𝜃 → 𝜏 ) )
3		bj-exlimd.maj	⊢ ( 𝜓 → ( 𝜒 → 𝜃 ) )
4	1 3	sylg	⊢ ( 𝜑 → ∀ 𝑥 ( 𝜒 → 𝜃 ) )
5		bj-exlimg	⊢ ( ( ∃ 𝑥 𝜃 → 𝜏 ) → ( ∀ 𝑥 ( 𝜒 → 𝜃 ) → ( ∃ 𝑥 𝜒 → 𝜏 ) ) )
6	2 4 5	sylc	⊢ ( 𝜑 → ( ∃ 𝑥 𝜒 → 𝜏 ) )