Theorem bj-sb56 32639

Description: Proof of sb56 2150 from Tarski, ax-10 2019 (modal5) and bj-ax12 32634. (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-sb56	⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-sb56

Step	Hyp	Ref	Expression
1		bj-ax12 32634	. . . 4 ⊢ ∀𝑥(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2		pm3.31 461	. . . . 5 ⊢ ((𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) → ((𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3	2	aleximi 1759	. . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥∀𝑥(𝑥 = 𝑦 → 𝜑)))
4	1, 3	ax-mp 5	. . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥∀𝑥(𝑥 = 𝑦 → 𝜑))
5		bj-modal5e 32636	. . 3 ⊢ (∃𝑥∀𝑥(𝑥 = 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
6	4, 5	syl 17	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
7		equs4v 1930	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
8	6, 7	impbii 199	1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  bj-dfssb2  32640