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Metamath Proof Explorer

Theorem dral1-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 dral1 using ax-c11 . (Contributed by NM, 24-Nov-1994) (New usage is discouraged.)

Ref Expression
Hypothesis dral1-o.1 ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion dral1-o ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝜑 ↔ ∀ 𝑦 𝜓 ) )

Proof

Step Hyp Ref Expression
1 dral1-o.1 ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
2 hbae-o ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑥 ∀ 𝑥 𝑥 = 𝑦 )
3 1 biimpd ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
4 2 3 alimdh ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) )
5 ax-c11 ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝜓 → ∀ 𝑦 𝜓 ) )
6 4 5 syld ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) )
7 hbae-o ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑦 ∀ 𝑥 𝑥 = 𝑦 )
8 1 biimprd ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜓 → 𝜑 ) )
9 7 8 alimdh ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑦 𝜓 → ∀ 𝑦 𝜑 ) )
10 ax-c11 ⊢ ( ∀ 𝑦 𝑦 = 𝑥 → ( ∀ 𝑦 𝜑 → ∀ 𝑥 𝜑 ) )
11 10 aecoms-o ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑦 𝜑 → ∀ 𝑥 𝜑 ) )
12 9 11 syld ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑦 𝜓 → ∀ 𝑥 𝜑 ) )
13 6 12 impbid ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝜑 ↔ ∀ 𝑦 𝜓 ) )