Theorem equsexvw 2007

Description: Version of equsexv 2276 with a disjoint variable condition, and of equsex 2423 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  2261  cleljustab  2718  axsepgfromrep  5230  dfid3  5529  opeliunxp  5698  opeliun2xp  5699  imai  6040  coi1  6228  opabex3d  7918  opabex3rd  7919  opabex3  7920  fsplit  8067  mapsnend  8983  elirrv  9512  dfac5lem1  10045  dfac5lem3  10047  dffix2  36085  sscoid  36093  elfuns  36095  pmapglb  40216  polval2N  40352  tfsconcat0i  43773