Theorem equs5

Description: Lemma used in proofs of substitution properties. If there is a disjoint variable condition on x , y , then sbalex can be used instead; if y is not free in ph , then equs45f can be used. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 14-May-1993) (Revised by BJ, 1-Oct-2018) (New usage is discouraged.)

	Ref	Expression
Assertion	equs5	$$⊢\neg \forall xx=y\to \left(\exists x\left(x=y\wedge \phi \right)\leftrightarrow \forall x\left(x=y\to \phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		nfna1	$$⊢Ⅎx\neg \forall xx=y$$
2		nfa1	$$⊢Ⅎx\forall x\left(x=y\to \phi \right)$$
3		axc15	$$⊢\neg \forall xx=y\to \left(x=y\to \left(\phi \to \forall x\left(x=y\to \phi \right)\right)\right)$$
4	3	impd	$$⊢\neg \forall xx=y\to \left(\left(x=y\wedge \phi \right)\to \forall x\left(x=y\to \phi \right)\right)$$
5	1 2 4	exlimd	$$⊢\neg \forall xx=y\to \left(\exists x\left(x=y\wedge \phi \right)\to \forall x\left(x=y\to \phi \right)\right)$$
6		equs4	$$⊢\forall x\left(x=y\to \phi \right)\to \exists x\left(x=y\wedge \phi \right)$$
7	5 6	impbid1	$$⊢\neg \forall xx=y\to \left(\exists x\left(x=y\wedge \phi \right)\leftrightarrow \forall x\left(x=y\to \phi \right)\right)$$