ac6mapd

Description: Axiom of choice equivalent, deduction form. (Contributed by Thierry Arnoux, 13-Oct-2025)

	Ref	Expression
Hypotheses	ac6mapd.1	$⊢ y = f (x) \to (ψ \leftrightarrow χ)$
	ac6mapd.2	$⊢ φ \to A \in V$
	ac6mapd.3	$⊢ φ \to B \in W$
	ac6mapd.4	$⊢ (φ \land x \in A) \to \exists y \in B ψ$
Assertion	ac6mapd	$⊢ φ \to \exists f \in (B^{A}) \forall x \in A χ$

Step	Hyp	Ref	Expression
1		ac6mapd.1	$⊢ y = f (x) \to (ψ \leftrightarrow χ)$
2		ac6mapd.2	$⊢ φ \to A \in V$
3		ac6mapd.3	$⊢ φ \to B \in W$
4		ac6mapd.4	$⊢ (φ \land x \in A) \to \exists y \in B ψ$
5	4	ralrimiva	$⊢ φ \to \forall x \in A \exists y \in B ψ$
6	1	ac6sg	$⊢ A \in V \to (\forall x \in A \exists y \in B ψ \to \exists f (f : A ⟶ B \land \forall x \in A χ))$
7	2 5 6	sylc	$⊢ φ \to \exists f (f : A ⟶ B \land \forall x \in A χ)$
8	3 2	elmapd	$⊢ φ \to (f \in (B^{A}) \leftrightarrow f : A ⟶ B)$
9	8	biimprd	$⊢ φ \to (f : A ⟶ B \to f \in (B^{A}))$
10	9	anim1d	$⊢ φ \to ((f : A ⟶ B \land \forall x \in A χ) \to (f \in (B^{A}) \land \forall x \in A χ))$
11	10	eximdv	$⊢ φ \to (\exists f (f : A ⟶ B \land \forall x \in A χ) \to \exists f (f \in (B^{A}) \land \forall x \in A χ))$
12	7 11	mpd	$⊢ φ \to \exists f (f \in (B^{A}) \land \forall x \in A χ)$
13		df-rex	$⊢ \exists f \in (B^{A}) \forall x \in A χ \leftrightarrow \exists f (f \in (B^{A}) \land \forall x \in A χ)$
14	12 13	sylibr	$⊢ φ \to \exists f \in (B^{A}) \forall x \in A χ$

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