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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ceqsalgALT | Structured version Visualization version GIF version |
Description: Alternate proof of bj-ceqsalg 32072. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bj-ceqsalg.1 | ⊢ Ⅎ𝑥𝜓 |
bj-ceqsalg.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
bj-ceqsalgALT | ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-ceqsalg.1 | . 2 ⊢ Ⅎ𝑥𝜓 | |
2 | bj-ceqsalg.2 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 2 | ax-gen 1713 | . 2 ⊢ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
4 | bj-ceqsalt 32069 | . 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) | |
5 | 1, 3, 4 | mp3an12 1406 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 = wceq 1475 Ⅎwnf 1699 ∈ wcel 1977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-clel 2606 |
This theorem is referenced by: (None) |
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