Theorem wl-lem-moexsb 38106

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 2192	. . 3 ⊢ Ⅎ𝑥∀𝑥(𝜑 → 𝑥 = 𝑧)
2		nfs1v 2197	. . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
3		sp 2225	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → (𝜑 → 𝑥 = 𝑧))
4		ax12v2 2221	. . . . 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 2125	. . . 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 2260	. 2 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → (∃𝑥𝜑 → [𝑧 / 𝑥]𝜑))
9		spsbe 2122	. 2 ⊢ ([𝑧 / 𝑥]𝜑 → ∃𝑥𝜑)
10	8, 9	impbid1 228	1 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → (∃𝑥𝜑 ↔ [𝑧 / 𝑥]𝜑))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1565  ∃wex 1806  [wsb 2097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-10 2182  ax-12 2219

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1807  df-nf 1811  df-sb 2098

This theorem is referenced by: (None)