Theorem sbh 1776

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 1766	. . . 4 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
2		sbh.1	. . . . 5 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	19.41h 1685	. . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ (∃𝑥 𝑥 = 𝑦 ∧ 𝜑))
4	1, 3	sylib 122	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 → (∃𝑥 𝑥 = 𝑦 ∧ 𝜑))
5	4	simprd 114	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → 𝜑)
6		stdpc4 1775	. . 3 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
7	2, 6	syl 14	. 2 ⊢ (𝜑 → [𝑦 / 𝑥]𝜑)
8	5, 7	impbii 126	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1351  ∃wex 1492  [wsb 1762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-i9 1530  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-sb 1763

This theorem is referenced by:  sbf  1777  sb6x  1779  nfs1f  1780  hbs1f  1781  sbid2h  1849  sblimv  1894  sbrim  1956  sbrbif  1962  elsb1  2155  elsb2  2156