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Metamath Proof Explorer

Theorem dral2-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 dral2 using ax-c11 . (Contributed by NM, 27-Feb-2005) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	dral2-o.1	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
	Assertion	dral2-o	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑧 𝜑 ↔ ∀ 𝑧 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		dral2-o.1	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
2		hbae-o	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑧 ∀ 𝑥 𝑥 = 𝑦 )
3	2 1	albidh	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑧 𝜑 ↔ ∀ 𝑧 𝜓 ) )