Theorem bj-spcimdvv

Description: Remove from spcimdv dependency on ax-7 , ax-8 , ax-10 , ax-11 , ax-12 ax-13 , ax-ext , df-cleq , df-clab (and df-nfc , df-v , df-or , df-tru , df-nf ) at the price of adding a disjoint variable condition on x , B (but in usages, x is typically a dummy, hence fresh, variable). For the version without this disjoint variable condition, see bj-spcimdv . (Contributed by BJ, 3-Nov-2021) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	bj-spcimdvv.1	$$⊢\phi \to A\in B$$
	bj-spcimdvv.2	$$⊢\left(\phi \wedge x=A\right)\to \left(\psi \to \chi \right)$$
Assertion	bj-spcimdvv	$$⊢\phi \to \left(\forall x\psi \to \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-spcimdvv.1	$$⊢\phi \to A\in B$$
2		bj-spcimdvv.2	$$⊢\left(\phi \wedge x=A\right)\to \left(\psi \to \chi \right)$$
3	2	ex	$$⊢\phi \to \left(x=A\to \left(\psi \to \chi \right)\right)$$
4	3	alrimiv	$$⊢\phi \to \forall x\left(x=A\to \left(\psi \to \chi \right)\right)$$
5		elissetv	$$⊢A\in B\to \exists xx=A$$
6		exim	$$⊢\forall x\left(x=A\to \left(\psi \to \chi \right)\right)\to \left(\exists xx=A\to \exists x\left(\psi \to \chi \right)\right)$$
7	5 6	syl5	$$⊢\forall x\left(x=A\to \left(\psi \to \chi \right)\right)\to \left(A\in B\to \exists x\left(\psi \to \chi \right)\right)$$
8		19.36v	$$⊢\exists x\left(\psi \to \chi \right)\leftrightarrow \left(\forall x\psi \to \chi \right)$$
9	7 8	imbitrdi	$$⊢\forall x\left(x=A\to \left(\psi \to \chi \right)\right)\to \left(A\in B\to \left(\forall x\psi \to \chi \right)\right)$$
10	4 1 9	sylc	$$⊢\phi \to \left(\forall x\psi \to \chi \right)$$