Theorem sb5f1 917

Description: Equivalence for substitution. Similar to Theorem 6.2 of [Quine] p. 40.

Hypothesis

Ref	Expression
sb5f1.1	⊢ (φ → ∀xφ)

Assertion

Ref	Expression
sb5f1	⊢ (φ ↔ ∀x(x = y → [x / y]φ))

Proof of Theorem sb5f1

Step	Hyp	Ref	Expression
1		sb5f1.1	. . 3 ⊢ (φ → ∀xφ)
2		sbequ1 863	. . . . 5 ⊢ (y = x → (φ → [x / y]φ))
3	2	eqcoms 813	. . . 4 ⊢ (x = y → (φ → [x / y]φ))
4	3	com12 13	. . 3 ⊢ (φ → (x = y → [x / y]φ))
5	1, 4	19.21ai 740	. 2 ⊢ (φ → ∀x(x = y → [x / y]φ))
6		sb2 859	. . . 4 ⊢ (∀x(x = y → [x / y]φ) → [y / x][x / y]φ)
7		sbco 910	. . . 4 ⊢ ([y / x][x / y]φ ↔ [y / x]φ)
8	6, 7	sylib 173	. . 3 ⊢ (∀x(x = y → [x / y]φ) → [y / x]φ)
9	1	sbf 870	. . 3 ⊢ ([y / x]φ ↔ φ)
10	8, 9	sylib 173	. 2 ⊢ (∀x(x = y → [x / y]φ) → φ)
11	5, 10	impbi 139	1 ⊢ (φ ↔ ∀x(x = y → [x / y]φ))

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672   = weq 797  [wsb 852

This theorem is referenced by:  eu1 1019

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853

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