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Metamath Proof Explorer

Theorem mosub

Description: "At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995)

		Ref	Expression
	Hypothesis	mosub.1	$⊢ \exists^{*} x φ$
	Assertion	mosub	$⊢ \exists^{*} x \exists y (y = A \land φ)$

Step	Hyp	Ref	Expression
1		mosub.1	$⊢ \exists^{*} x φ$
2		moeq	$⊢ \exists^{*} y y = A$
3	1	ax-gen	$⊢ \forall y \exists^{*} x φ$
4		moexexvw	$⊢ (\exists^{} y y = A \land \forall y \exists^{} x φ) \to \exists^{*} x \exists y (y = A \land φ)$
5	2 3 4	mp2an	$⊢ \exists^{*} x \exists y (y = A \land φ)$