Theorem sbh 1637

Description: Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 17-Oct-2004.)

Hypothesis

Ref	Expression
sbh.1	⊢ (φ → ∀xφ)

Assertion

Ref	Expression
sbh	⊢ ([y / x]φ ↔ φ)

Proof of Theorem sbh

Step	Hyp	Ref	Expression
1		sb1 1627	. . . 4 ⊢ ([y / x]φ → ∃x(x = y ∧ φ))
2		sbh.1	. . . . 5 ⊢ (φ → ∀xφ)
3	2	19.41h 1553	. . . 4 ⊢ (∃x(x = y ∧ φ) ↔ (∃x x = y ∧ φ))
4	1, 3	sylib 127	. . 3 ⊢ ([y / x]φ → (∃x x = y ∧ φ))
5	4	simprd 107	. 2 ⊢ ([y / x]φ → φ)
6		stdpc4 1636	. . 3 ⊢ (∀xφ → [y / x]φ)
7	2, 6	syl 14	. 2 ⊢ (φ → [y / x]φ)
8	5, 7	impbii 117	1 ⊢ ([y / x]φ ↔ φ)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1224  ∃wex 1358  [wsb 1623

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1312  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-4 1377  ax-i9 1400  ax-ial 1405

This theorem depends on definitions:  df-bi 110  df-sb 1624

This theorem is referenced by:  sbf  1638  sb6x  1640  nfs1f  1641  hbs1f  1642  sbid2h  1707  sblimv  1752  sbrim  1808  sbrbif  1814  elsb3  1830  elsb4  1831