Theorem sbidm 1252

Description: An idempotent law for substitution.

Assertion

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

Proof of Theorem sbidm

Step	Hyp	Ref	Expression
1		sbequ12 1179	. . . 4 ⊢ (x = y → ([y / x]φ ↔ [y / x][y / x]φ))
2	1	bicomd 520	. . 3 ⊢ (x = y → ([y / x][y / x]φ ↔ [y / x]φ))
3	2	a4s 982	. 2 ⊢ (∀x x = y → ([y / x][y / x]φ ↔ [y / x]φ))
4		hbnae 1145	. . 3 ⊢ (¬ ∀x x = y → ∀x ¬ ∀x x = y)
5		hbsb2 1225	. . 3 ⊢ (¬ ∀x x = y → ([y / x]φ → ∀x[y / x]φ))
6		pm4.2i 171	. . . 4 ⊢ (x = y → ([y / x]φ ↔ [y / x]φ))
7	6	a1i 8	. . 3 ⊢ (¬ ∀x x = y → (x = y → ([y / x]φ ↔ [y / x]φ)))
8	4, 5, 7	sbied 1193	. 2 ⊢ (¬ ∀x x = y → ([y / x][y / x]φ ↔ [y / x]φ))
9	3, 8	pm2.61i 126	1 ⊢ ([y / x][y / x]φ ↔ [y / x]φ)

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146  ∀wal 952  [wsbc 1168

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-10 964  ax-12 966  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-11o 1216

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170

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