Theorem ralxpes

Description: A version of ralxp with explicit substitution. (Contributed by Scott Fenton, 21-Aug-2024)

	Ref	Expression
Assertion	ralxpes	$$⊢\forall x\in \left(A\times B\right)\underset{\dot{}}{[}{1}^{st}\left(x\right)/y\underset{\dot{}}{]}\underset{\dot{}}{[}{2}^{nd}\left(x\right)/z\underset{\dot{}}{]}\phi \leftrightarrow \forall y\in A\forall z\in B\phi$$

Proof

Step	Hyp	Ref	Expression
1		nfsbc1v	$$⊢Ⅎy\underset{\dot{}}{[}{1}^{st}\left(x\right)/y\underset{\dot{}}{]}\underset{\dot{}}{[}{2}^{nd}\left(x\right)/z\underset{\dot{}}{]}\phi$$
2		nfcv	$$⊢\underset{\_}{Ⅎ}z{1}^{st}\left(x\right)$$
3		nfsbc1v	$$⊢Ⅎz\underset{\dot{}}{[}{2}^{nd}\left(x\right)/z\underset{\dot{}}{]}\phi$$
4	2 3	nfsbcw	$$⊢Ⅎz\underset{\dot{}}{[}{1}^{st}\left(x\right)/y\underset{\dot{}}{]}\underset{\dot{}}{[}{2}^{nd}\left(x\right)/z\underset{\dot{}}{]}\phi$$
5		nfv	$$⊢Ⅎx\phi$$
6		sbcopeq1a	$$⊢x=⟨y,z⟩\to \left(\underset{\dot{}}{[}{1}^{st}\left(x\right)/y\underset{\dot{}}{]}\underset{\dot{}}{[}{2}^{nd}\left(x\right)/z\underset{\dot{}}{]}\phi \leftrightarrow \phi \right)$$
7	1 4 5 6	ralxpf	$$⊢\forall x\in \left(A\times B\right)\underset{\dot{}}{[}{1}^{st}\left(x\right)/y\underset{\dot{}}{]}\underset{\dot{}}{[}{2}^{nd}\left(x\right)/z\underset{\dot{}}{]}\phi \leftrightarrow \forall y\in A\forall z\in B\phi$$