Theorem drex1 2327

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.)

Hypothesis

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

Assertion

Ref	Expression
drex1	⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓))

Proof of Theorem drex1

Step	Hyp	Ref	Expression
1		dral1.1	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2	1	notbid 308	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
3	2	dral1 2325	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 ¬ 𝜑 ↔ ∀𝑦 ¬ 𝜓))
4	3	notbid 308	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 ¬ 𝜑 ↔ ¬ ∀𝑦 ¬ 𝜓))
5		df-ex 1705	. 2 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
6		df-ex 1705	. 2 ⊢ (∃𝑦𝜓 ↔ ¬ ∀𝑦 ¬ 𝜓)
7	4, 5, 6	3bitr4g 303	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196  ∀wal 1481  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  exdistrf  2333  drsb1  2377  eujustALT  2473  copsexg  4956  dfid3  5025  dropab1  38651  dropab2  38652  e2ebind  38779  e2ebindVD  39148  e2ebindALT  39165