Theorem dral2-o 2181

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 dral2 1966 using ax-10o 2139. (Contributed by NM, 27-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
dral2-o.1	⊢ (∀x x = y → (φ ↔ ψ))

Assertion

Ref	Expression
dral2-o	⊢ (∀x x = y → (∀zφ ↔ ∀zψ))

Proof of Theorem dral2-o

Step	Hyp	Ref	Expression
1		hbae-o 2153	. 2 ⊢ (∀x x = y → ∀z∀x x = y)
2		dral2-o.1	. 2 ⊢ (∀x x = y → (φ ↔ ψ))
3	1, 2	albidh 1590	1 ⊢ (∀x x = y → (∀zφ ↔ ∀zψ))

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:  ax11eq  2193  ax11el  2194  ax11indalem  2197  ax11inda2ALT  2198