Theorem bj-ceqsalt 36844

Description: Remove from ceqsalt 3523 dependency on ax-ext 2711 (and on df-cleq 2732 and df-v 3490). Note: this is not doable with ceqsralt 3524 (or ceqsralv 3531), which uses eleq1 2832, but the same dependence removal is possible for ceqsalg 3525, ceqsal 3527, ceqsalv 3529, cgsexg 3536, cgsex2g 3537, cgsex4g 3538, ceqsex 3540, ceqsexv 3542, ceqsex2 3547, ceqsex2v 3548, ceqsex3v 3549, ceqsex4v 3550, ceqsex6v 3551, ceqsex8v 3552, gencbvex 3553 (after changing 𝐴 = 𝑦 to 𝑦 = 𝐴), gencbvex2 3554, gencbval 3555, vtoclgft 3564 (it uses Ⅎ, whose justification nfcjust 2894 does not use ax-ext 2711) and several other vtocl* theorems (see for instance bj-vtoclg1f 36876). See also bj-ceqsaltv 36845. (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 2826	. . 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 36842	. 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 1087  ∀wal 1535   = wceq 1537  ∃wex 1777  Ⅎwnf 1781   ∈ wcel 2108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-clel 2819

This theorem is referenced by:  bj-ceqsalgALT  36848