equsalv

Description: An equivalence related to implicit substitution. Version of equsal with a disjoint variable condition, which does not require ax-13 . See equsalvw for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsexv . (Contributed by NM, 2-Jun-1993) (Revised by BJ, 31-May-2019)

	Ref	Expression
Hypotheses	equsalv.nf	⊢ Ⅎ 𝑥 𝜓
	equsalv.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	equsalv	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ↔ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		equsalv.nf	⊢ Ⅎ 𝑥 𝜓
2		equsalv.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
3	1	19.23	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜓 ) ↔ ( ∃ 𝑥 𝑥 = 𝑦 → 𝜓 ) )
4	2	pm5.74i	⊢ ( ( 𝑥 = 𝑦 → 𝜑 ) ↔ ( 𝑥 = 𝑦 → 𝜓 ) )
5	4	albii	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜓 ) )
6		ax6ev	⊢ ∃ 𝑥 𝑥 = 𝑦
7	6	a1bi	⊢ ( 𝜓 ↔ ( ∃ 𝑥 𝑥 = 𝑦 → 𝜓 ) )
8	3 5 7	3bitr4i	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ↔ 𝜓 )

Metamath Proof Explorer

Theorem equsalv

Proof