Theorem qremexd-P6 823

Description: Existential Quantifier Removal Theorem (deductive form).

Hypothesis

Ref	Expression
qremexd-P6.1	⊢ (𝛾 → Ⅎ𝑥𝜑)

Assertion

Ref	Expression
qremexd-P6	⊢ (𝛾 → (∃𝑥𝜑 ↔ 𝜑))

Proof of Theorem qremexd-P6

Step	Hyp	Ref	Expression
1		qremexd-P6.1	. . . 4 ⊢ (𝛾 → Ⅎ𝑥𝜑)
2		dfnfreealt-P6 683	. . . 4 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
3	1, 2	subimr2-P4.RC 543	. . 3 ⊢ (𝛾 → (∃𝑥𝜑 → ∀𝑥𝜑))
4		spec-P6 719	. . . 4 ⊢ (∀𝑥𝜑 → 𝜑)
5	4	rcp-NDIMP0addall 207	. . 3 ⊢ (𝛾 → (∀𝑥𝜑 → 𝜑))
6	3, 5	syl-P3.24 259	. 2 ⊢ (𝛾 → (∃𝑥𝜑 → 𝜑))
7		exi-P6 718	. . 3 ⊢ (𝜑 → ∃𝑥𝜑)
8	7	rcp-NDIMP0addall 207	. 2 ⊢ (𝛾 → (𝜑 → ∃𝑥𝜑))
9	6, 8	ndbii-P3.13 178	1 ⊢ (𝛾 → (∃𝑥𝜑 ↔ 𝜑))

Syntax hints:  ∀wff-forall 8   → wff-imp 10   ↔ wff-bi 104  ∃wff-exists 595  Ⅎwff-nfree 681

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14  ax-GEN 15  ax-L4 16  ax-L5 17  ax-L6 18  ax-L7 19  ax-L12 29

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-or-D2.3 145  df-true-D2.4 155  df-rcp-AND3 161  df-exists-D5.1 596  df-nfree-D6.1 682

This theorem is referenced by:  exiad-P6  824