Theorem wl-lem-moexsb 38256

Description: The antecedent ∀𝑥(𝜑 → 𝑥 = 𝑧) relates to ∃*𝑥𝜑, but is better suited for usage in proofs. Note that no distinct variable restriction is placed on 𝜑.

This theorem provides a basic working step in proving theorems about ∃* or ∃!. (Contributed by Wolf Lammen, 3-Oct-2019.)

Assertion

Ref	Expression
wl-lem-moexsb	⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → (∃𝑥𝜑 ↔ [𝑧 / 𝑥]𝜑))

Distinct variable group:   𝑥,𝑧

Allowed substitution hints:   𝜑(𝑥, 𝑧)

Proof of Theorem wl-lem-moexsb

Step	Hyp	Ref	Expression
1		nfa1 2189	. . 3 ⊢ Ⅎ𝑥∀𝑥(𝜑 → 𝑥 = 𝑧)
2		nfs1v 2194	. . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
3		sp 2222	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → (𝜑 → 𝑥 = 𝑧))
4		ax12v2 2218	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))
5	3, 4	syli 40	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))
6		sb6 2122	. . . 4 ⊢ ([𝑧 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑧 → 𝜑))
7	5, 6	imbitrrdi 255	. . 3 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → (𝜑 → [𝑧 / 𝑥]𝜑))
8	1, 2, 7	exlimd 2257	. 2 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → (∃𝑥𝜑 → [𝑧 / 𝑥]𝜑))
9		spsbe 2119	. 2 ⊢ ([𝑧 / 𝑥]𝜑 → ∃𝑥𝜑)
10	8, 9	impbid1 228	1 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → (∃𝑥𝜑 ↔ [𝑧 / 𝑥]𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  ∃wex 1812  [wsb 2099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-sb 2100

This theorem is used by: (None)