Theorem drex2

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of Megill p. 448 (p. 16 of preprint). Usage of this theorem is discouraged because it depends on ax-13 . Usage of exbidv is preferred, which requires fewer axioms. (Contributed by NM, 27-Feb-2005) (New usage is discouraged.)

	Ref	Expression
Hypothesis	dral1.1	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	drex2	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∃ 𝑧 𝜑 ↔ ∃ 𝑧 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		dral1.1	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
2		nfae	⊢ Ⅎ 𝑧 ∀ 𝑥 𝑥 = 𝑦
3	2 1	exbid	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∃ 𝑧 𝜑 ↔ ∃ 𝑧 𝜓 ) )