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Metamath Proof Explorer

Theorem sb5f

Description: Equivalence for substitution when y is not free in ph . The implication "to the right" is sb1 and does not require the nonfreeness hypothesis. Theorem sb5 replaces the nonfreeness hypothesis with a disjoint variable condition on x , y and requires fewer axioms. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 5-Aug-1993) (Revised by Mario Carneiro, 4-Oct-2016) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	sb6f.1	\|- F/ y ph
	Assertion	sb5f	\|- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) )

Proof

Step	Hyp	Ref	Expression
1		sb6f.1	\|- F/ y ph
2	1	sb6f	\|- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )
3	1	equs45f	\|- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) )
4	2 3	bitr4i	\|- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) )