Theorem drex1 1770

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 1696	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑥∀𝑥 𝑥 = 𝑦)
2		drex1.1	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3		ax-4 1487	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦)
4	3	biantrurd 303	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜓 ↔ (𝑥 = 𝑦 ∧ 𝜓)))
5	2, 4	bitr2d 188	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝑥 = 𝑦 ∧ 𝜓) ↔ 𝜑))
6	1, 5	exbidh 1593	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) ↔ ∃𝑥𝜑))
7		ax11e 1768	. . . 4 ⊢ (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) → ∃𝑦𝜓))
8	7	sps 1517	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) → ∃𝑦𝜓))
9	6, 8	sylbird 169	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 → ∃𝑦𝜓))
10		hbae 1696	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦∀𝑥 𝑥 = 𝑦)
11		equcomi 1680	. . . . . . 7 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)
12	11	sps 1517	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑦 = 𝑥)
13	12	biantrurd 303	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ (𝑦 = 𝑥 ∧ 𝜑)))
14	13, 2	bitr3d 189	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝑦 = 𝑥 ∧ 𝜑) ↔ 𝜓))
15	10, 14	exbidh 1593	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑦(𝑦 = 𝑥 ∧ 𝜑) ↔ ∃𝑦𝜓))
16		ax11e 1768	. . . . 5 ⊢ (𝑦 = 𝑥 → (∃𝑦(𝑦 = 𝑥 ∧ 𝜑) → ∃𝑥𝜑))
17	16	sps 1517	. . . 4 ⊢ (∀𝑦 𝑦 = 𝑥 → (∃𝑦(𝑦 = 𝑥 ∧ 𝜑) → ∃𝑥𝜑))
18	17	alequcoms 1496	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑦(𝑦 = 𝑥 ∧ 𝜑) → ∃𝑥𝜑))
19	15, 18	sylbird 169	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑦𝜓 → ∃𝑥𝜑))
20	9, 19	impbid 128	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1329   = wceq 1331  ∃wex 1468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  drsb1  1771  exdistrfor  1772  copsexg  4161