Theorem 2sb6rf 1341

Description: Reversed double substitution.

Hypotheses

Ref	Expression
2sb5rf.1	⊢ (φ → ∀zφ)
2sb5rf.2	⊢ (φ → ∀wφ)

Assertion

Ref	Expression
2sb6rf	⊢ (φ ↔ ∀z∀w((z = x ⋀ w = y) → [z / x][w / y]φ))

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

Proof of Theorem 2sb6rf

Step	Hyp	Ref	Expression
1		2sb5rf.1	. . 3 ⊢ (φ → ∀zφ)
2	1	sb6rf 1262	. 2 ⊢ (φ ↔ ∀z(z = x → [z / x]φ))
3		19.21v 1287	. . . 4 ⊢ (∀w(z = x → (w = y → [w / y][z / x]φ)) ↔ (z = x → ∀w(w = y → [w / y][z / x]φ)))
4		sbcom2 1336	. . . . . . 7 ⊢ ([z / x][w / y]φ ↔ [w / y][z / x]φ)
5	4	imbi2i 185	. . . . . 6 ⊢ (((z = x ⋀ w = y) → [z / x][w / y]φ) ↔ ((z = x ⋀ w = y) → [w / y][z / x]φ))
6		impexp 347	. . . . . 6 ⊢ (((z = x ⋀ w = y) → [w / y][z / x]φ) ↔ (z = x → (w = y → [w / y][z / x]φ)))
7	5, 6	bitr 173	. . . . 5 ⊢ (((z = x ⋀ w = y) → [z / x][w / y]φ) ↔ (z = x → (w = y → [w / y][z / x]φ)))
8	7	albii 1001	. . . 4 ⊢ (∀w((z = x ⋀ w = y) → [z / x][w / y]φ) ↔ ∀w(z = x → (w = y → [w / y][z / x]φ)))
9		2sb5rf.2	. . . . . . 7 ⊢ (φ → ∀wφ)
10	9	hbsb 1335	. . . . . 6 ⊢ ([z / x]φ → ∀w[z / x]φ)
11	10	sb6rf 1262	. . . . 5 ⊢ ([z / x]φ ↔ ∀w(w = y → [w / y][z / x]φ))
12	11	imbi2i 185	. . . 4 ⊢ ((z = x → [z / x]φ) ↔ (z = x → ∀w(w = y → [w / y][z / x]φ)))
13	3, 8, 12	3bitr4r 184	. . 3 ⊢ ((z = x → [z / x]φ) ↔ ∀w((z = x ⋀ w = y) → [z / x][w / y]φ))
14	13	albii 1001	. 2 ⊢ (∀z(z = x → [z / x]φ) ↔ ∀z∀w((z = x ⋀ w = y) → [z / x][w / y]φ))
15	2, 14	bitr 173	1 ⊢ (φ ↔ ∀z∀w((z = x ⋀ w = y) → [z / x][w / y]φ))

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

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174

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