Theorem equs4 1148

Description: Lemma used in proofs of substitution properties.

Assertion

Ref	Expression
equs4	⊢ (∀x(x = y → φ) → ∃x(x = y ⋀ φ))

Proof of Theorem equs4

Step	Hyp	Ref	Expression
1		pm3.27 323	. . . . . . . 8 ⊢ ((∀x(x = y → φ) ⋀ x = y) → x = y)
2		ax-4 971	. . . . . . . . 9 ⊢ (∀x(x = y → φ) → (x = y → φ))
3	2	imp 350	. . . . . . . 8 ⊢ ((∀x(x = y → φ) ⋀ x = y) → φ)
4	1, 3	jc 138	. . . . . . 7 ⊢ ((∀x(x = y → φ) ⋀ x = y) → ¬ (x = y → ¬ φ))
5		ax-4 971	. . . . . . 7 ⊢ (∀x(x = y → ¬ φ) → (x = y → ¬ φ))
6	4, 5	nsyl 116	. . . . . 6 ⊢ ((∀x(x = y → φ) ⋀ x = y) → ¬ ∀x(x = y → ¬ φ))
7	6	ex 373	. . . . 5 ⊢ (∀x(x = y → φ) → (x = y → ¬ ∀x(x = y → ¬ φ)))
8		hbn1 1013	. . . . 5 ⊢ (¬ ∀x(x = y → ¬ φ) → ∀x ¬ ∀x(x = y → ¬ φ))
9	7, 8	syl6 22	. . . 4 ⊢ (∀x(x = y → φ) → (x = y → ∀x ¬ ∀x(x = y → ¬ φ)))
10	9	a5i 987	. . 3 ⊢ (∀x(x = y → φ) → ∀x(x = y → ∀x ¬ ∀x(x = y → ¬ φ)))
11		ax-9o 1121	. . 3 ⊢ (∀x(x = y → ∀x ¬ ∀x(x = y → ¬ φ)) → ¬ ∀x(x = y → ¬ φ))
12	10, 11	syl 10	. 2 ⊢ (∀x(x = y → φ) → ¬ ∀x(x = y → ¬ φ))
13		equs3 1147	. 2 ⊢ (∃x(x = y ⋀ φ) ↔ ¬ ∀x(x = y → ¬ φ))
14	12, 13	sylibr 200	1 ⊢ (∀x(x = y → φ) → ∃x(x = y ⋀ φ))

Syntax hints:  ¬ wn 2   → wi 3   ⋀ wa 223  ∀wal 952   = wceq 954  ∃wex 978

This theorem is referenced by:  sb2 1175  equs45f 1198

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 961  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121

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

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