Theorem dral1 1153

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint).

Hypothesis

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

Assertion

Ref	Expression
dral1	⊢ (∀x x = y → (∀xφ ↔ ∀yψ))

Proof of Theorem dral1

Step	Hyp	Ref	Expression
1		hbae 1144	. . . 4 ⊢ (∀x x = y → ∀x∀x x = y)
2		dral1.1	. . . . 5 ⊢ (∀x x = y → (φ ↔ ψ))
3	2	biimpd 153	. . . 4 ⊢ (∀x x = y → (φ → ψ))
4	1, 3	19.20d 995	. . 3 ⊢ (∀x x = y → (∀xφ → ∀xψ))
5		ax-10o 1139	. . 3 ⊢ (∀x x = y → (∀xψ → ∀yψ))
6	4, 5	syld 27	. 2 ⊢ (∀x x = y → (∀xφ → ∀yψ))
7		hbae 1144	. . . 4 ⊢ (∀x x = y → ∀y∀x x = y)
8	2	biimprd 154	. . . 4 ⊢ (∀x x = y → (ψ → φ))
9	7, 8	19.20d 995	. . 3 ⊢ (∀x x = y → (∀yψ → ∀yφ))
10		ax-10o 1139	. . . 4 ⊢ (∀y y = x → (∀yφ → ∀xφ))
11	10	alequcoms 1142	. . 3 ⊢ (∀x x = y → (∀yφ → ∀xφ))
12	9, 11	syld 27	. 2 ⊢ (∀x x = y → (∀yψ → ∀xφ))
13	6, 12	impbid 515	1 ⊢ (∀x x = y → (∀xφ ↔ ∀yψ))

Syntax hints:   → wi 3   ↔ wb 146  ∀wal 953   = wceq 955

This theorem is referenced by:  drex1 1155  ax11 1218  hbsb4 1247  sb9i 1262  a16g 1275  ax11indalem 1367  ax11inda2ALT 1368  ralcom2 1774  axpownd 4936

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-10 965  ax-12 967  ax-4 972  ax-5o 974  ax-10o 1139

This theorem depends on definitions:  df-bi 147  df-an 225

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