Theorem sbid2 911

Description: An identity law for substitution.

Hypothesis

Ref	Expression
sbid2.1	⊢ (φ → ∀xφ)

Assertion

Ref	Expression
sbid2	⊢ ([y / x][x / y]φ ↔ φ)

Proof of Theorem sbid2

Step	Hyp	Ref	Expression
1		sbco 910	. 2 ⊢ ([y / x][x / y]φ ↔ [y / x]φ)
2		sbid2.1	. . 3 ⊢ (φ → ∀xφ)
3	2	sbf 870	. 2 ⊢ ([y / x]φ ↔ φ)
4	1, 3	bitr 151	1 ⊢ ([y / x][x / y]φ ↔ φ)

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

This theorem is referenced by:  sbco2 913  sb8 918  sbid2v 993

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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