Theorem rspsbc2

Description: rspsbc with two quantifying variables. This proof is rspsbc2VD automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	rspsbc2	$$⊢A\in B\to \left(C\in D\to \left(\forall x\in B\forall y\in D\phi \to \underset{\dot{}}{[}C/y\underset{\dot{}}{]}\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		idd	$$⊢A\in B\to \left(C\in D\to C\in D\right)$$
2		rspsbc	$$⊢A\in B\to \left(\forall x\in B\forall y\in D\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\forall y\in D\phi \right)$$
3	2	a1d	$$⊢A\in B\to \left(C\in D\to \left(\forall x\in B\forall y\in D\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\forall y\in D\phi \right)\right)$$
4		sbcralg	$$⊢A\in B\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\forall y\in D\phi \leftrightarrow \forall y\in D\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)$$
5	4	biimpd	$$⊢A\in B\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\forall y\in D\phi \to \forall y\in D\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)$$
6	3 5	syl6d	$$⊢A\in B\to \left(C\in D\to \left(\forall x\in B\forall y\in D\phi \to \forall y\in D\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)\right)$$
7		rspsbc	$$⊢C\in D\to \left(\forall y\in D\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}C/y\underset{\dot{}}{]}\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)$$
8	1 6 7	syl10	$$⊢A\in B\to \left(C\in D\to \left(\forall x\in B\forall y\in D\phi \to \underset{\dot{}}{[}C/y\underset{\dot{}}{]}\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)\right)$$