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Metamath Proof Explorer

Theorem 19.21t

Description: Closed form of Theorem 19.21 of Margaris p. 90, see 19.21 . (Contributed by NM, 27-May-1997) (Revised by Mario Carneiro, 24-Sep-2016) (Proof shortened by Wolf Lammen, 3-Jan-2018) df-nf changed. (Revised by Wolf Lammen, 11-Sep-2021) (Proof shortened by BJ, 3-Nov-2021)

		Ref	Expression
	Assertion	19.21t	\|- ( F/ x ph -> ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) )

Proof

Step	Hyp	Ref	Expression
1		19.38a	\|- ( F/ x ph -> ( ( E. x ph -> A. x ps ) <-> A. x ( ph -> ps ) ) )
2		19.9t	\|- ( F/ x ph -> ( E. x ph <-> ph ) )
3	2	imbi1d	\|- ( F/ x ph -> ( ( E. x ph -> A. x ps ) <-> ( ph -> A. x ps ) ) )
4	1 3	bitr3d	\|- ( F/ x ph -> ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) )