Theorem bj-spcimdv

Description: Remove from spcimdv dependency on ax-9 , ax-10 , ax-11 , ax-13 , ax-ext , df-cleq (and df-nfc , df-v , df-or , df-tru , df-nf ). For an even more economical version, see bj-spcimdvv . (Contributed by BJ, 30-Nov-2020) (Proof modification is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		bj-spcimdv.1	$$⊢\phi \to A\in B$$
2		bj-spcimdv.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		elisset	$$⊢A\in B\to \exists xx=A$$
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	$$⊢\phi \to \left(\forall x\psi \to \chi \right)$$