Theorem drex1 1661

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.) (Revised by NM, 3-Feb-2015.)

Hypothesis

Ref	Expression
drex1.1	⊢ (∀x x = y → (φ ↔ ψ))

Assertion

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

Proof of Theorem drex1

Step	Hyp	Ref	Expression
1		hbae 1588	. . . 4 ⊢ (∀x x = y → ∀x∀x x = y)
2		drex1.1	. . . . 5 ⊢ (∀x x = y → (φ ↔ ψ))
3		ax-4 1381	. . . . . 6 ⊢ (∀x x = y → x = y)
4	3	biantrurd 289	. . . . 5 ⊢ (∀x x = y → (ψ ↔ (x = y ∧ ψ)))
5	2, 4	bitr2d 178	. . . 4 ⊢ (∀x x = y → ((x = y ∧ ψ) ↔ φ))
6	1, 5	exbidh 1487	. . 3 ⊢ (∀x x = y → (∃x(x = y ∧ ψ) ↔ ∃xφ))
7		ax11e 1659	. . . 4 ⊢ (x = y → (∃x(x = y ∧ ψ) → ∃yψ))
8	7	sps 1412	. . 3 ⊢ (∀x x = y → (∃x(x = y ∧ ψ) → ∃yψ))
9	6, 8	sylbird 159	. 2 ⊢ (∀x x = y → (∃xφ → ∃yψ))
10		hbae 1588	. . . 4 ⊢ (∀x x = y → ∀y∀x x = y)
11		equcomi 1574	. . . . . . 7 ⊢ (x = y → y = x)
12	11	sps 1412	. . . . . 6 ⊢ (∀x x = y → y = x)
13	12	biantrurd 289	. . . . 5 ⊢ (∀x x = y → (φ ↔ (y = x ∧ φ)))
14	13, 2	bitr3d 179	. . . 4 ⊢ (∀x x = y → ((y = x ∧ φ) ↔ ψ))
15	10, 14	exbidh 1487	. . 3 ⊢ (∀x x = y → (∃y(y = x ∧ φ) ↔ ∃yψ))
16		ax11e 1659	. . . . 5 ⊢ (y = x → (∃y(y = x ∧ φ) → ∃xφ))
17	16	sps 1412	. . . 4 ⊢ (∀y y = x → (∃y(y = x ∧ φ) → ∃xφ))
18	17	alequcoms 1390	. . 3 ⊢ (∀x x = y → (∃y(y = x ∧ φ) → ∃xφ))
19	15, 18	sylbird 159	. 2 ⊢ (∀x x = y → (∃yψ → ∃xφ))
20	9, 19	impbid 120	1 ⊢ (∀x x = y → (∃xφ ↔ ∃yψ))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1226   = wceq 1228  ∃wex 1362

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  drsb1  1662  exdistrfor  1663  copsexg  3955