Theorem sb6rf 1867

Description: Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
sb5rf.1	⊢ (𝜑 → ∀𝑦𝜑)

Assertion

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

Proof of Theorem sb6rf

Step	Hyp	Ref	Expression
1		sb5rf.1	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
2		sbequ1 1782	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
3	2	equcoms 1722	. . . 4 ⊢ (𝑦 = 𝑥 → (𝜑 → [𝑦 / 𝑥]𝜑))
4	3	com12 30	. . 3 ⊢ (𝜑 → (𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))
5	1, 4	alrimih 1483	. 2 ⊢ (𝜑 → ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))
6		sb2 1781	. . 3 ⊢ (∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑) → [𝑥 / 𝑦][𝑦 / 𝑥]𝜑)
7	1	sbid2h 1863	. . 3 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ 𝜑)
8	6, 7	sylib 122	. 2 ⊢ (∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑) → 𝜑)
9	5, 8	impbii 126	1 ⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1362  [wsb 1776

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-sb 1777

This theorem is referenced by:  2sb6rf  2009  eu1  2070