Theorem wl-lem-moexsb 34806

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 2155	. . 3 ⊢ Ⅎ𝑥∀𝑥(𝜑 → 𝑥 = 𝑧)
2		nfs1v 2160	. . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
3		sp 2182	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → (𝜑 → 𝑥 = 𝑧))
4		ax12v2 2179	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))
5	3, 4	syli 39	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))
6		sb6 2093	. . . 4 ⊢ ([𝑧 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑧 → 𝜑))
7	5, 6	syl6ibr 254	. . 3 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → (𝜑 → [𝑧 / 𝑥]𝜑))
8	1, 2, 7	exlimd 2218	. 2 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → (∃𝑥𝜑 → [𝑧 / 𝑥]𝜑))
9		spsbe 2088	. 2 ⊢ ([𝑧 / 𝑥]𝜑 → ∃𝑥𝜑)
10	8, 9	impbid1 227	1 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → (∃𝑥𝜑 ↔ [𝑧 / 𝑥]𝜑))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1535  ∃wex 1780  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070

This theorem is referenced by: (None)