Theorem equsexvw 2007

Description: Version of equsexv 2275 with a disjoint variable condition, and of equsex 2421 with two disjoint variable conditions, which requires fewer axioms. See also the dual form equsalvw 2006. (Contributed by BJ, 31-May-2019.) (Proof shortened by Wolf Lammen, 23-Oct-2023.)

Hypothesis

Ref	Expression
equsalvw.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
equsexvw	⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓)

Distinct variable groups:   𝑥,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)

Proof of Theorem equsexvw

Step	Hyp	Ref	Expression
1		alinexa 1845	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → ¬ 𝜑) ↔ ¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
2		equsalvw.1	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	2	notbid 318	. . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
4	3	equsalvw 2006	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → ¬ 𝜑) ↔ ¬ 𝜓)
5	1, 4	bitr3i 277	. 2 ⊢ (¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ¬ 𝜓)
6	5	con4bii 321	1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782

This theorem is referenced by:  equvinv  2031  cleljust  2123  sbelx  2260  cleljustab  2716  axsepgfromrep  5218  dfid3  5518  opeliunxp  5687  opeliun2xp  5688  imai  6028  coi1  6216  opabex3d  7907  opabex3rd  7908  opabex3  7909  fsplit  8056  mapsnend  8972  elirrv  9501  dfac5lem1  10034  dfac5lem3  10036  dffix2  36073  sscoid  36081  elfuns  36083  pmapglb  40204  polval2N  40340  tfsconcat0i  43761