Theorem sb8e 919

Description: Substitution of variable in existential quantifier.

Hypothesis

Ref	Expression
sb8e.1	⊢ (φ → ∀yφ)

Assertion

Ref	Expression
sb8e	⊢ (∃xφ ↔ ∃y[y / x]φ)

Proof of Theorem sb8e

Step	Hyp	Ref	Expression
1		sb8e.1	. . . . . 6 ⊢ (φ → ∀yφ)
2	1	hbne 699	. . . . 5 ⊢ (¬ φ → ∀y ¬ φ)
3	2	sb8 918	. . . 4 ⊢ (∀x ¬ φ ↔ ∀y[y / x] ¬ φ)
4		sbn 882	. . . . 5 ⊢ ([y / x] ¬ φ ↔ ¬ [y / x]φ)
5	4	bial 695	. . . 4 ⊢ (∀y[y / x] ¬ φ ↔ ∀y ¬ [y / x]φ)
6	3, 5	bitr 151	. . 3 ⊢ (∀x ¬ φ ↔ ∀y ¬ [y / x]φ)
7	6	negbii 162	. 2 ⊢ (¬ ∀x ¬ φ ↔ ¬ ∀y ¬ [y / x]φ)
8		df-ex 679	. 2 ⊢ (∃xφ ↔ ¬ ∀x ¬ φ)
9		df-ex 679	. 2 ⊢ (∃y[y / x]φ ↔ ¬ ∀y ¬ [y / x]φ)
10	7, 8, 9	3bitr4 158	1 ⊢ (∃xφ ↔ ∃y[y / x]φ)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127  ∀wal 672  ∃wex 678  [wsb 852

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

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

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