Theorem sbex 998

Description: Move existential quantifier in and out of substitution.

Assertion

Ref	Expression
sbex	⊢ ([z / y]∃xφ ↔ ∃x[z / y]φ)

Distinct variable group(s):   x,y   x,z

Proof of Theorem sbex

Step	Hyp	Ref	Expression
1		sbn 882	. . 3 ⊢ ([z / y] ¬ ∀x ¬ φ ↔ ¬ [z / y]∀x ¬ φ)
2		sbal 997	. . . . 5 ⊢ ([z / y]∀x ¬ φ ↔ ∀x[z / y] ¬ φ)
3		sbn 882	. . . . . 6 ⊢ ([z / y] ¬ φ ↔ ¬ [z / y]φ)
4	3	bial 695	. . . . 5 ⊢ (∀x[z / y] ¬ φ ↔ ∀x ¬ [z / y]φ)
5	2, 4	bitr 151	. . . 4 ⊢ ([z / y]∀x ¬ φ ↔ ∀x ¬ [z / y]φ)
6	5	negbii 162	. . 3 ⊢ (¬ [z / y]∀x ¬ φ ↔ ¬ ∀x ¬ [z / y]φ)
7	1, 6	bitr 151	. 2 ⊢ ([z / y] ¬ ∀x ¬ φ ↔ ¬ ∀x ¬ [z / y]φ)
8		df-ex 679	. . 3 ⊢ (∃xφ ↔ ¬ ∀x ¬ φ)
9	8	bisb 855	. 2 ⊢ ([z / y]∃xφ ↔ [z / y] ¬ ∀x ¬ φ)
10		df-ex 679	. 2 ⊢ (∃x[z / y]φ ↔ ¬ ∀x ¬ [z / y]φ)
11	7, 9, 10	3bitr4 158	1 ⊢ ([z / y]∃xφ ↔ ∃x[z / y]φ)

Syntax hints:  ¬ wn 1   ↔ wb 127  ∀wal 672  ∃wex 678  [wsb 852

This theorem is referenced by:  sbabel 1189  sbcex 1465

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  ax-16 922

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853

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