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Mirrors > Home > MPE Home > Th. List > elequ2g | Structured version Visualization version GIF version |
Description: A form of elequ2 2125 with a universal quantifier. Its converse is the axiom of extensionality ax-ext 2708. (Contributed by BJ, 3-Oct-2019.) |
Ref | Expression |
---|---|
elequ2g | ⊢ (𝑥 = 𝑦 → ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elequ2 2125 | . 2 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)) | |
2 | 1 | alrimiv 1935 | 1 ⊢ (𝑥 = 𝑦 → ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-9 2120 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 |
This theorem is referenced by: axextb 2711 |
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