Metamath Proof Explorer


Theorem drex1

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 . Use the weaker drex1v if possible. (Contributed by NM, 27-Feb-2005) (New usage is discouraged.)

Ref Expression
Hypothesis dral1.1 ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑𝜓 ) )
Assertion drex1 ( ∀ 𝑥 𝑥 = 𝑦 → ( ∃ 𝑥 𝜑 ↔ ∃ 𝑦 𝜓 ) )

Proof

Step Hyp Ref Expression
1 dral1.1 ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑𝜓 ) )
2 1 notbid ( ∀ 𝑥 𝑥 = 𝑦 → ( ¬ 𝜑 ↔ ¬ 𝜓 ) )
3 2 dral1 ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 ¬ 𝜑 ↔ ∀ 𝑦 ¬ 𝜓 ) )
4 3 notbid ( ∀ 𝑥 𝑥 = 𝑦 → ( ¬ ∀ 𝑥 ¬ 𝜑 ↔ ¬ ∀ 𝑦 ¬ 𝜓 ) )
5 df-ex ( ∃ 𝑥 𝜑 ↔ ¬ ∀ 𝑥 ¬ 𝜑 )
6 df-ex ( ∃ 𝑦 𝜓 ↔ ¬ ∀ 𝑦 ¬ 𝜓 )
7 4 5 6 3bitr4g ( ∀ 𝑥 𝑥 = 𝑦 → ( ∃ 𝑥 𝜑 ↔ ∃ 𝑦 𝜓 ) )