Theorem drex1 2441

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 2372. Use the weaker drex1v 2369 if possible. (Contributed by NM, 27-Feb-2005.) (New usage is discouraged.)

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 317	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
3	2	dral1 2439	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 ¬ 𝜑 ↔ ∀𝑦 ¬ 𝜓))
4	3	notbid 317	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 ¬ 𝜑 ↔ ¬ ∀𝑦 ¬ 𝜓))
5		df-ex 1784	. 2 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
6		df-ex 1784	. 2 ⊢ (∃𝑦𝜓 ↔ ¬ ∀𝑦 ¬ 𝜓)
7	4, 5, 6	3bitr4g 313	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205  ∀wal 1537  ∃wex 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-12 2173  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1784  df-nf 1788

This theorem is referenced by:  exdistrf  2447  drsb1  2499  eujustALT  2572  copsexg  5399  dfid3  5483  dropab1  41954  dropab2  41955  e2ebind  42072  e2ebindVD  42421  e2ebindALT  42438