Theorem 2sb6 1338

Description: Equivalence for double substitution.

Assertion

Ref	Expression
2sb6	⊢ ([z / x][w / y]φ ↔ ∀x∀y((x = z ⋀ y = w) → φ))

Distinct variable groups:   x,y,z   y,w

Proof of Theorem 2sb6

Step	Hyp	Ref	Expression
1		sb6 1269	. 2 ⊢ ([z / x][w / y]φ ↔ ∀x(x = z → [w / y]φ))
2		19.21v 1287	. . . 4 ⊢ (∀y(x = z → (y = w → φ)) ↔ (x = z → ∀y(y = w → φ)))
3		impexp 347	. . . . 5 ⊢ (((x = z ⋀ y = w) → φ) ↔ (x = z → (y = w → φ)))
4	3	albii 1001	. . . 4 ⊢ (∀y((x = z ⋀ y = w) → φ) ↔ ∀y(x = z → (y = w → φ)))
5		sb6 1269	. . . . 5 ⊢ ([w / y]φ ↔ ∀y(y = w → φ))
6	5	imbi2i 185	. . . 4 ⊢ ((x = z → [w / y]φ) ↔ (x = z → ∀y(y = w → φ)))
7	2, 4, 6	3bitr4r 184	. . 3 ⊢ ((x = z → [w / y]φ) ↔ ∀y((x = z ⋀ y = w) → φ))
8	7	albii 1001	. 2 ⊢ (∀x(x = z → [w / y]φ) ↔ ∀x∀y((x = z ⋀ y = w) → φ))
9	1, 8	bitr 173	1 ⊢ ([z / x][w / y]φ ↔ ∀x∀y((x = z ⋀ y = w) → φ))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 956   = wceq 958  [wsbc 1172

This theorem is referenced by:  2eu6 1457

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-16 1212  ax-11o 1220

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174

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