Theorem bj-ceqsalt 36880

Description: Remove from ceqsalt 3470 dependency on ax-ext 2701 (and on df-cleq 2721 and df-v 3438). Note: this is not doable with ceqsralt 3471 (or ceqsralv 3477), which uses eleq1 2816, but the same dependence removal is possible for ceqsalg 3472, ceqsal 3474, ceqsalv 3476, cgsexg 3481, cgsex2g 3482, cgsex4g 3483, ceqsex 3485, ceqsexv 3487, ceqsex2 3490, ceqsex2v 3491, ceqsex3v 3492, ceqsex4v 3493, ceqsex6v 3494, ceqsex8v 3495, gencbvex 3496 (after changing 𝐴 = 𝑦 to 𝑦 = 𝐴), gencbvex2 3497, gencbval 3498, vtoclgft 3507 (it uses Ⅎ, whose justification nfcjust 2877 does not use ax-ext 2701) and several other vtocl* theorems (see for instance bj-vtoclg1f 36912). See also bj-ceqsaltv 36881. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-ceqsalt	⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem bj-ceqsalt

Step	Hyp	Ref	Expression
1		elisset 2810	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2	1	3anim3i 1154	. 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∃𝑥 𝑥 = 𝐴))
3		bj-ceqsalt0 36878	. 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∃𝑥 𝑥 = 𝐴) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
4	2, 3	syl 17	1 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086  ∀wal 1538   = wceq 1540  ∃wex 1779  Ⅎwnf 1783   ∈ wcel 2109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-clel 2803

This theorem is referenced by:  bj-ceqsalgALT  36884