Theorem dral2-o 39049

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 2440 using ax-c11 39006. (Contributed by NM, 27-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

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

Assertion

Ref	Expression
dral2-o	⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓))

Proof of Theorem dral2-o

Step	Hyp	Ref	Expression
1		hbae-o 39022	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑥 𝑥 = 𝑦)
2		dral2-o.1	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	1, 2	albidh 1867	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-11 2162  ax-c5 39002  ax-c4 39003  ax-c7 39004  ax-c10 39005  ax-c11 39006  ax-c9 39009

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781

This theorem is referenced by:  ax12eq  39060  ax12el  39061  ax12indalem  39064  ax12inda2ALT  39065