Theorem sbc3an

Description: Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006) (Revised by NM, 17-Aug-2018)

	Ref	Expression
Assertion	sbc3an	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \wedge \psi \wedge \chi \right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \wedge \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \wedge \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		df-3an	$$⊢\left(\phi \wedge \psi \wedge \chi \right)\leftrightarrow \left(\left(\phi \wedge \psi \right)\wedge \chi \right)$$
2	1	sbcbii	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \wedge \psi \wedge \chi \right)\leftrightarrow \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\left(\phi \wedge \psi \right)\wedge \chi \right)$$
3		sbcan	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\left(\phi \wedge \psi \right)\wedge \chi \right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \wedge \psi \right)\wedge \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)$$
4		sbcan	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \wedge \psi \right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \wedge \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)$$
5	4	anbi1i	$$⊢\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \wedge \psi \right)\wedge \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\leftrightarrow \left(\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \wedge \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)\wedge \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)$$
6	2 3 5	3bitri	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \wedge \psi \wedge \chi \right)\leftrightarrow \left(\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \wedge \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)\wedge \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)$$
7		df-3an	$$⊢\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \wedge \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \wedge \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\leftrightarrow \left(\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \wedge \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)\wedge \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)$$
8	6 7	bitr4i	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \wedge \psi \wedge \chi \right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \wedge \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \wedge \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)$$