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Metamath Proof Explorer

Theorem bj-sylge

Description: Dual statement of sylg (the final "e" in the label stands for "existential (version of sylg )". Variant of exlimih . (Contributed by BJ, 25-Dec-2023)

	Ref	Expression
Hypotheses	bj-sylge.nf	⊢ ( ∃ 𝑥 𝜑 → 𝜓 )
	bj-sylge.maj	⊢ ( 𝜒 → 𝜑 )
Assertion	bj-sylge	⊢ ( ∃ 𝑥 𝜒 → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		bj-sylge.nf	⊢ ( ∃ 𝑥 𝜑 → 𝜓 )
2		bj-sylge.maj	⊢ ( 𝜒 → 𝜑 )
3	2	eximi	⊢ ( ∃ 𝑥 𝜒 → ∃ 𝑥 𝜑 )
4	3 1	syl	⊢ ( ∃ 𝑥 𝜒 → 𝜓 )