Theorem dral1-o 2154

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). Version of dral1 1965 using ax-10o 2139. (Contributed by NM, 24-Nov-1994.) (New usage is discouraged.)

Hypothesis

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

Assertion

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

Proof of Theorem dral1-o

Step	Hyp	Ref	Expression
1		hbae-o 2153	. . . 4 ⊢ (∀x x = y → ∀x∀x x = y)
2		dral1-o.1	. . . . 5 ⊢ (∀x x = y → (φ ↔ ψ))
3	2	biimpd 198	. . . 4 ⊢ (∀x x = y → (φ → ψ))
4	1, 3	alimdh 1563	. . 3 ⊢ (∀x x = y → (∀xφ → ∀xψ))
5		ax-10o 2139	. . 3 ⊢ (∀x x = y → (∀xψ → ∀yψ))
6	4, 5	syld 40	. 2 ⊢ (∀x x = y → (∀xφ → ∀yψ))
7		hbae-o 2153	. . . 4 ⊢ (∀x x = y → ∀y∀x x = y)
8	2	biimprd 214	. . . 4 ⊢ (∀x x = y → (ψ → φ))
9	7, 8	alimdh 1563	. . 3 ⊢ (∀x x = y → (∀yψ → ∀yφ))
10		ax-10o 2139	. . . 4 ⊢ (∀y y = x → (∀yφ → ∀xφ))
11	10	aecoms-o 2152	. . 3 ⊢ (∀x x = y → (∀yφ → ∀xφ))
12	9, 11	syld 40	. 2 ⊢ (∀x x = y → (∀yψ → ∀xφ))
13	6, 12	impbid 183	1 ⊢ (∀x x = y → (∀xφ ↔ ∀yψ))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540

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-7 1734  ax-4 2135  ax-5o 2136  ax-6o 2137  ax-10o 2139  ax-12o 2142

This theorem depends on definitions:  df-bi 177  df-ex 1542

This theorem is referenced by:  ax11  2155  a16g-o  2186  ax11indalem  2197  ax11inda2ALT  2198