Theorem wl-dfcleq.basic 38015

Description: This theorem is a conservative extension of ax-ext 2737 to classes, with no hypotheses. It is not complete, since ax-8 2147 can be derived (see in-ax8 36597) via alpha-renaming.

Although unsuitable for general use, it is adequate for the development of theorems unaffected by alpha-renaming, including:

1. Theorems with no bound variables in the hypotheses or conclusion (see eqriv 2762).

2. Theorems using the same bound variable throughout (see abbib 2834).

3. Theorems with distinct bound variables arising only through implicit substitution (see eqabbw 2838).

Remark: the proof uses axextb 2740 to prove the hypothesis of df-cleq 2757 that is a degenerate instance, but it could be proved also from minimal propositional calculus and { ax-gen 1818, equid 2035 }. (Contributed by NM, 15-Sep-1993.) (Revised by BJ, 24-Jun-2019.)

Assertion

Ref	Expression
wl-dfcleq.basic	⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem wl-dfcleq.basic

Dummy variables 𝑦 𝑧 𝑡 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		axextb 2740	. 2 ⊢ (𝑦 = 𝑧 ↔ ∀𝑢(𝑢 ∈ 𝑦 ↔ 𝑢 ∈ 𝑧))
2		axextb 2740	. 2 ⊢ (𝑡 = 𝑡 ↔ ∀𝑣(𝑣 ∈ 𝑡 ↔ 𝑣 ∈ 𝑡))
3	1, 2	wl-df.cleq 38014	1 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Syntax hints:   ↔ wb 209  ∀wal 1561   = wceq 1563   ∈ wcel 2145

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803  df-cleq 2757

This theorem is referenced by:  wl-dfcleq.just  38016  wl-dfcleq  38020