Theorem drex1 1967

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 285	. . . 4 ⊢ (∀x x = y → (¬ φ ↔ ¬ ψ))
3	2	dral1 1965	. . 3 ⊢ (∀x x = y → (∀x ¬ φ ↔ ∀y ¬ ψ))
4	3	notbid 285	. 2 ⊢ (∀x x = y → (¬ ∀x ¬ φ ↔ ¬ ∀y ¬ ψ))
5		df-ex 1542	. 2 ⊢ (∃xφ ↔ ¬ ∀x ¬ φ)
6		df-ex 1542	. 2 ⊢ (∃yψ ↔ ¬ ∀y ¬ ψ)
7	4, 5, 6	3bitr4g 279	1 ⊢ (∀x x = y → (∃xφ ↔ ∃yψ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  exdistrf  1971  drsb1  2022  eujustALT  2207  copsexg  4608  dfid3  4769