Theorem dral1-o

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of Megill p. 448 (p. 16 of preprint). Version of dral1 using ax-c11 . (Contributed by NM, 24-Nov-1994) (New usage is discouraged.)

	Ref	Expression
Hypothesis	dral1-o.1	$$⊢\forall xx=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	dral1-o	$$⊢\forall xx=y\to \left(\forall x\phi \leftrightarrow \forall y\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		dral1-o.1	$$⊢\forall xx=y\to \left(\phi \leftrightarrow \psi \right)$$
2		hbae-o	$$⊢\forall xx=y\to \forall x\forall xx=y$$
3	1	biimpd	$$⊢\forall xx=y\to \left(\phi \to \psi \right)$$
4	2 3	alimdh	$$⊢\forall xx=y\to \left(\forall x\phi \to \forall x\psi \right)$$
5		ax-c11	$$⊢\forall xx=y\to \left(\forall x\psi \to \forall y\psi \right)$$
6	4 5	syld	$$⊢\forall xx=y\to \left(\forall x\phi \to \forall y\psi \right)$$
7		hbae-o	$$⊢\forall xx=y\to \forall y\forall xx=y$$
8	1	biimprd	$$⊢\forall xx=y\to \left(\psi \to \phi \right)$$
9	7 8	alimdh	$$⊢\forall xx=y\to \left(\forall y\psi \to \forall y\phi \right)$$
10		ax-c11	$$⊢\forall yy=x\to \left(\forall y\phi \to \forall x\phi \right)$$
11	10	aecoms-o	$$⊢\forall xx=y\to \left(\forall y\phi \to \forall x\phi \right)$$
12	9 11	syld	$$⊢\forall xx=y\to \left(\forall y\psi \to \forall x\phi \right)$$
13	6 12	impbid	$$⊢\forall xx=y\to \left(\forall x\phi \leftrightarrow \forall y\psi \right)$$